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%I A008292 #719 Nov 06 2024 04:46:39
%S A008292 1,1,1,1,4,1,1,11,11,1,1,26,66,26,1,1,57,302,302,57,1,1,120,1191,2416,
%T A008292 1191,120,1,1,247,4293,15619,15619,4293,247,1,1,502,14608,88234,
%U A008292 156190,88234,14608,502,1,1,1013,47840,455192,1310354,1310354,455192,47840,1013,1
%N A008292 Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows.
%C A008292 The indexing used here follows that given in the classic books by Riordan and Comtet. For two other versions see A173018 and A123125. - _N. J. A. Sloane_, Nov 21 2010
%C A008292 Coefficients of Eulerian polynomials. Number of permutations of n objects with k-1 rises. Number of increasing rooted trees with n+1 nodes and k leaves.
%C A008292 T(n,k) = number of permutations of [n] with k runs. T(n,k) = number of permutations of [n] requiring k readings (see the Knuth reference). T(n,k) = number of permutations of [n] having k distinct entries in its inversion table. - _Emeric Deutsch_, Jun 09 2004
%C A008292 T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004
%C A008292 Subtriangle for k>=1 and n>=1 of triangle A123125. - _Philippe Deléham_, Oct 22 2006
%C A008292 T(n,k)/n! also represents the n-dimensional volume of the portion of the n-dimensional hypercube cut by the (n-1)-dimensional hyperplanes x_1 + x_2 + ... x_n = k, x_1 + x_2 + ... x_n = k-1; or, equivalently, it represents the probability that the sum of n independent random variables with uniform distribution between 0 and 1 is between k-1 and k. - Stefano Zunino, Oct 25 2006
%C A008292 [E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and [-P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials and P(n,t) are the polynomials in A131758. - _Tom Copeland_, Sep 30 2007
%C A008292 From _Tom Copeland_, Oct 07 2008: (Start)
%C A008292 G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1*x + (2+t)*x^2/2! + (6+6*t+t^2)*x^3/3! + ... gives row polynomials for A090582, the reverse f-polynomials for the permutohedra (see A019538).
%C A008292 G(x,t-1) = 1 + 1*x + (1+t)*x^2/2! + (1+4*t+t^2)*x^3/3! + ... gives row polynomials for A008292, the h-polynomials for permutohedra (Postnikov et al.).
%C A008292 G((t+1)*x, -1/(t+1)) = 1 + (1+t)*x + (1+3*t+2*t^2)*x^2/2! + ... gives row polynomials for A028246.
%C A008292 (End)
%C A008292 A subexceedant function f on [n] is a map f:[n] -> [n] such that 1 <= f(i) <= i for all i, 1 <= i <= n. T(n,k) equals the number of subexceedant functions f of [n] such that the image of f has cardinality k [Mantaci & Rakotondrajao]. Example T(3,2) = 4: if we identify a subexceedant function f with the word f(1)f(2)...f(n) then the subexceedant functions on [3] are 111, 112, 113, 121, 122 and 123 and four of these functions have an image set of cardinality 2. - _Peter Bala_, Oct 21 2008
%C A008292 Further to the comments of _Tom Copeland_ above, the n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type A_(n-1). The corresponding f-vectors are the rows of A019538. For example, 1 + 4*x + x^2 = y^2 + 6*y + 6 and 1 + 11*x + 11*x^2 + x^3 = y^3 + 14*y^2 + 36*y + 24, where x = y + 1, give [1,6,6] and [1,14,36,24] as the third and fourth rows of A019538. The Hilbert transform of this triangle (see A145905 for the definition) is A047969. See A060187 for the triangle of Eulerian numbers of type B (the h-vectors of the simplicial complexes dual to permutohedra of type B). See A066094 for the array of h-vectors of type D. For tables of restricted Eulerian numbers see A144696 - A144699. - _Peter Bala_, Oct 26 2008
%C A008292 For a natural refinement of A008292 with connections to compositional inversion and iterated derivatives, see A145271. - _Tom Copeland_, Nov 06 2008
%C A008292 The polynomials E(z,n) = numerator(Sum_{k>=1} (-1)^(n+1)*k^n*z^(k-1)) for n >=1 lead directly to the triangle of Eulerian numbers. - _Johannes W. Meijer_, May 24 2009
%C A008292 From Walther Janous (walther.janous(AT)tirol.com), Nov 01 2009: (Start)
%C A008292 The (Eulerian) polynomials e(n,x) = Sum_{k=0..n-1} T(n,k+1)*x^k turn out to be also the numerators of the closed-form expressions of the infinite sums:
%C A008292 S(p,x) = Sum_{j>=0} (j+1)^p*x^j, that is
%C A008292 S(p,x) = e(p,x)/(1-x)^(p+1), whenever |x| < 1 and p is a positive integer.
%C A008292 (Note the inconsistent use of T(n,k) in the section listing the formula section. I adhere tacitly to the first one.) (End)
%C A008292 If n is an odd prime, then all numbers of the (n-2)-th and (n-1)-th rows are in the progression k*n+1. - _Vladimir Shevelev_, Jul 01 2011
%C A008292 The Eulerian triangle is an element of the formula for the r-th successive summation of Sum_{k=1..n} k^j which appears to be Sum_{k=1..n} T(j,k-1) * binomial(j-k+n+r, j+r). - _Gary Detlefs_, Nov 11 2011
%C A008292 Li and Wong show that T(n,k) counts the combinatorially inequivalent star polygons with n+1 vertices and sum of angles (2*k-n-1)*Pi. An equivalent formulation is: define the total sign change S(p) of a permutation p in the symmetric group S_n to be equal to Sum_{i=1..n} sign(p(i)-p(i+1)), where we take p(n+1) = p(1). T(n,k) gives the number of permutations q in S_(n+1) with q(1) = 1 and S(q) = 2*k-n-1. For example, T(3,2) = 4 since in S_4 the permutations (1243), (1324), (1342) and (1423) have total sign change 0. - _Peter Bala_, Dec 27 2011
%C A008292 Xiong, Hall and Tsao refer to Riordan and mention that a traditional Eulerian number A(n,k) is the number of permutations of (1,2...n) with k weak exceedances. - _Susanne Wienand_, Aug 25 2014
%C A008292 Connections to algebraic geometry/topology and characteristic classes are discussed in the Buchstaber and Bunkova, the Copeland, the Hirzebruch, the Lenart and Zainoulline, the Losev and Manin, and the Sheppeard links; to the Grassmannian, in the Copeland, the Farber and Postnikov, the Sheppeard, and the Williams links; and to compositional inversion and differential operators, in the Copeland and the Parker links. - _Tom Copeland_, Oct 20 2015
%C A008292 The bivariate e.g.f. noted in the formulas is related to multiplying edges in certain graphs discussed in the Aluffi-Marcolli link. See p. 42. - _Tom Copeland_, Dec 18 2016
%C A008292 Distribution of left children in treeshelves is given by a shift of the Eulerian numbers. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. See A278677, A278678 or A278679 for more definitions and examples. - _Sergey Kirgizov_, Dec 24 2016
%C A008292 The row polynomial P(n, x) = Sum_{k=1..n} T(n, k)*x^k appears in the numerator of the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} P(n, x)/(1 - x)^(n+2) for n >= 0 (with 0^0=1). See also triangle A131689 with a Mar 31 2017 comment for a rewritten form. For the e.g.f see A028246 with a Mar 13 2017 comment. - _Wolfdieter Lang_, Mar 31 2017
%C A008292 For relations to Ehrhart polynomials, volumes of polytopes, polylogarithms, the Todd operator, and other special functions, polynomials, and sequences, see A131758 and the references therein. - _Tom Copeland_, Jun 20 2017
%C A008292 For relations to values of the Riemann zeta function at integral arguments, see A131758 and the Dupont reference. - _Tom Copeland_, Mar 19 2018
%C A008292 Normalized volumes of the hypersimplices, attributed to Laplace. (Cf. the De Loera et al. reference, p. 327.) - _Tom Copeland_, Jun 25 2018
%D A008292 Mohammad K. Azarian, Geometric Series, Problem 329, Mathematics and Computer Education, Vol. 30, No. 1, Winter 1996, p. 101. Solution published in Vol. 31, No. 2, Spring 1997, pp. 196-197.
%D A008292 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 106.
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%D A008292 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254; 2nd. ed., p. 268.[Worpitzky's identity (6.37)]
%D A008292 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1998, Vol. 3, p. 47 (exercise 5.1.4 Nr. 20) and p. 605 (solution).
%D A008292 Meng Li and Ron Goldman. "Limits of sums for binomial and Eulerian numbers and their associated distributions." Discrete Mathematics 343.7 (2020): 111870.
%D A008292 Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page http://math.ucsd.edu/~remmel/
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%D A008292 T. K. Petersen, Eulerian Numbers, Birkhauser, 2015.
%D A008292 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%D A008292 R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.
%D A008292 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Figure M3416, Academic Press, 1995.
%D A008292 H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973, see p. 208.
%D A008292 D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 101.
%H A008292 T. D. Noe, Rows 1 to 100 of triangle, flattened.
%H A008292 V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
%H A008292 Takashi Agoh, On Generalized Euler Numbers and Polynomials Related to Values of the Lerch Zeta Function, Integers (2020) Vol. 20, Article A5.
%H A008292 P. Aluffi and M. Marcolli, Feynman motives and deletion-contraction, arXiv:0907.3225 [math-ph], 2009.
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%H A008292 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
%H A008292 J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv:1307.5624 [math.CO], 2013.
%H A008292 Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
%H A008292 Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See p. 6.
%H A008292 Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5.
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%H A008292 Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
%H A008292 Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
%H A008292 Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
%H A008292 Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
%H A008292 D. Barsky, Analyse p-adique et suites classiques de nombres, Sem. Loth. Comb. B05b (1981) 1-21.
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%H A008292 Edward A. Bender, Central and local limit theorems applied to asymptotic enumeration Journal of Combinatorial Theory, Series A, 15(1) (1973), 91-111. See Example 5.3.
%H A008292 F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
%H A008292 V. Buchstaber and E. Bunkova, Elliptic formal group laws, integral Hirzebruch genera and Krichever genera, arXiv:1010.0944 [math-ph], 2010, p. 35.
%H A008292 Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018.
%H A008292 F. Cachazo, S. He, and E. Y. Yuan, Scattering in Three Dimensions from Rational Maps, arXiv:1306.2962 [hep-th], 2013.
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%H A008292 David Callan, Problem 498, The College Mathematics Journal, Vol. 24, No. 2 (Mar., 1993), pp. 183-190 (8 pages).
%H A008292 David Callan, Shi-Mei Ma, and Toufik Mansour, Some Combinatorial Arrays Related to the Lotka-Volterra System, Electronic Journal of Combinatorics, Volume 22, Issue 2 (2015), Paper #P2.22.
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%H A008292 Tom Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera, 2015.
%H A008292 Tom Copeland, Reciprocity and Umbral Witchcraft: An Eve with Stirling, Bernoulli, Archimedes, Euler, Laguerre, and Worpitzky, 2020.
%H A008292 J. A. De Loera, J. Rambau, and F. Santos, Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, Vol. 25, Springer-Verlag, 2010.
%H A008292 Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
%H A008292 J. Desarmenien and D. Foata, The signed Eulerian Numbers
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%H A008292 D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
%H A008292 B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths Thesis, Brandeis Univ., Aug. 2008.
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%H A008292 Joseph A. Farrow, A Monte Carlo Approach to the 4D Scattering Equations, arXiv:1806.02732 [hep-th], 2018.
%H A008292 FindStat - Combinatorial Statistic Finder, The number of descents of a permutation, The number of exceedances (also excedences) of a permutation, and The number of weak exceedances (also weak excedences) of a permutation
%H A008292 D. Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.
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%H A008292 Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432.
%H A008292 E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
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%H A008292 Jason Fulman, Gene B. Kim, Sangchul Lee, and T. Kyle Petersen, On the joint distribution of descents and signs of permutations, arXiv:1910.04258 [math.CO], 2019.
%H A008292 D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021.
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%H A008292 Ira Gessel, The Smith College diploma problem.
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%H A008292 Mats Granvik, Do these ratios of the Eulerian numbers converge to the logarithm of x?, Mathematics Stack Exchange, Dec 30 2014.
%H A008292 Jim Haglund and Mirko Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations.
%H A008292 Thomas Hameister, Sujit Rao, and Connor Simpson, Chow rings of matroids and atomistic lattices, research paper, University of Minnesota, 2017, also arXiv:1802.04241 [math.CO], 2018.
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%H A008292 Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom
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%H A008292 Lucas Kang, Investigation of Rule 73 as Case Study of Class 4 Long-Distance Cellular Automata, arXiv:1310.3311 [nlin.CG], 2013.
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%H A008292 M-H. Li and N-C. Wong, Sums of angles of star polygons and the Eulerian Numbers, Southeast Asian Bulletin of Mathematics 2004.
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%H A008292 Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012.
%H A008292 Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
%H A008292 Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
%H A008292 Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
%H A008292 Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, On certain combinatorial expansions of descent polynomials and the change of grammars, arXiv:1802.02861 [math.CO], 2018.
%H A008292 S.-M. Ma, T. Mansour, and M. Schork, Normal ordering problem and the extensions of the Stirling grammar, arXiv:1308.0169 [math.CO], 2013.
%H A008292 Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
%H A008292 Shi-Mei Ma and Hai-Na Wang, Enumeration of a dual set of Stirling permutations by their alternating runs, arXiv:1506.08716 [math.CO], 2015.
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%H A008292 J. Riordan, Review of Frankel (1950) [Annotated scanned copy]
%H A008292 J. Riordan, Triangular permutation numbers, Proc. Amer. Math. Soc. 2 (1951) 429-432, r(x,t).
%H A008292 D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
%H A008292 G. Rzadkowski, Two formulas for Successive Derivatives and Their Applications, JIS 12 (2009) 09.8.2.
%H A008292 G. Rzadkowski, An Analytic Approach to Special Numbers and Polynomials, J. Int. Seq. 18 (2015) 15.8.8.
%H A008292 Grzegorz Rzadkowski and M. Urlinska, A Generalization of the Eulerian Numbers, arXiv preprint arXiv:1612.06635 [math.CO], 2016-2017.
%H A008292 J. Sack and H. Ulfarsson, Refined inversion statistics on permutations, arXiv:1106.1995 [math.CO], 2011.
%H A008292 M. Sheppeard, Constructive motives and scattering 2013 (p. 41).
%H A008292 D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]
%H A008292 M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
%H A008292 R. Sprugnoli, Alternating Weighted Sums of Inverses of Binomial Coefficients, J. Integer Sequences, 15 (2012), #12.6.3.
%H A008292 Yidong Sun and Liting Zhai, Some properties of a class of refined Eulerian polynomials, arXiv:1810.07956 [math.CO], 2018.
%H A008292 S. Tanimoto, A study of Eulerian numbers by means of an operator on permutations, Europ. J. Combin., 24 (2003), 33-43.
%H A008292 Eric Weisstein's World of Mathematics, Eulerian Number and Euler's Number Triangle
%H A008292 Susanne Wienand, plots of exceedances for permutations of [4]
%H A008292 L. K. Williams, Enumeration of totally positive Grassmann cells, arXiv:math/0307271 [math.CO], 2003-2004.
%H A008292 Anthony James Wood, Nonequilibrium steady states from a random-walk perspective, Ph. D. Thesis, The University of Edinburgh (Scotland, UK 2019).
%H A008292 Anthony J. Wood, Richard A. Blythe, and Martin R. Evans, Combinatorial mappings of exclusion processes, arXiv:1908.00942 [cond-mat.stat-mech], 2019.
%H A008292 Tingyao Xiong, Jonathan I. Hall, and Hung-Ping Tsao, Combinatorial Interpretation of General Eulerian Numbers, Journal of Discrete Mathematics, (2014), Article ID 870596, 6 pages.
%H A008292 D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012.
%H A008292 Yifan Zhang and George Grossman, A Combinatorial Proof for the Generating Function of Powers of a Second-Order Recurrence Sequence, J. Int. Seq. 21 (2018), #18.3.3.
%H A008292 Index entries for sequences related to rooted trees
%H A008292 Index entries for "core" sequences
%F A008292 T(n, k) = k * T(n-1, k) + (n-k+1) * T(n-1, k-1), T(1, 1) = 1.
%F A008292 T(n, k) = Sum_{j=0..k} (-1)^j * (k-j)^n * binomial(n+1, j).
%F A008292 Row sums = n! = A000142(n) unless n=0. - _Michael Somos_, Mar 17 2011
%F A008292 E.g.f. A(x, q) = Sum_{n>0} (Sum_{k=1..n} T(n, k) * q^k) * x^n / n! = q * ( e^(q*x) - e^x ) / ( q*e^x - e^(q*x) ) satisfies dA / dx = (A + 1) * (A + q). - _Michael Somos_, Mar 17 2011
%F A008292 For a column listing, n-th term: T(c, n) = c^(n+c-1) + Sum_{i=1..c-1} (-1)^i/i!*(c-i)^(n+c-1)*Product_{j=1..i} (n+c+1-j). - _Randall L Rathbun_, Jan 23 2002
%F A008292 From John Robertson (jpr2718(AT)aol.com), Sep 02 2002: (Start)
%F A008292 Four characterizations of Eulerian numbers T(i, n):
%F A008292 1. T(0, n)=1 for n>=1, T(i, 1)=0 for i>=1, T(i, n) = (n-i)T(i-1, n-1) + (i+1)T(i, n-1).
%F A008292 2. T(i, n) = Sum_{j=0..i} (-1)^j*binomial(n+1,j)*(i-j+1)^n for n>=1, i>=0.
%F A008292 3. Let C_n be the unit cube in R^n with vertices (e_1, e_2, ..., e_n) where each e_i is 0 or 1 and all 2^n combinations are used. Then T(i, n)/n! is the volume of C_n between the hyperplanes x_1 + x_2 + ... + x_n = i and x_1 + x_2 + ... + x_n = i+1. Hence T(i, n)/n! is the probability that i <= X_1 + X_2 + ... + X_n < i+1 where the X_j are independent uniform [0, 1] distributions. - See Ehrenborg & Readdy reference.
%F A008292 4. Let f(i, n) = T(i, n)/n!. The f(i, n) are the unique coefficients so that (1/(r-1)^(n+1)) Sum_{i=0..n-1} f(i, n) r^{i+1} = Sum_{j>=0} (j^n)/(r^j) whenever n>=1 and abs(r)>1. (End)
%F A008292 O.g.f. for n-th row: (1-x)^(n+1)*polylog(-n, x)/x. - _Vladeta Jovovic_, Sep 02 2002
%F A008292 Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] (positive integers interspersed with 0's) where DELTA is Deléham's operator defined in A084938.
%F A008292 Sum_{k=1..n} T(n, k)*2^k = A000629(n). - _Philippe Deléham_, Jun 05 2004
%F A008292 From _Tom Copeland_, Oct 10 2007: (Start)
%F A008292 Bell_n(x) = Sum_{j=0..n} S2(n,j) * x^j = Sum_{j=0..n} E(n,j) * Lag(n,-x, j-n) = Sum_{j=0..n} (E(n,j)/n!) * (n!*Lag(n,-x, j-n)) = Sum_{j=0..n} E(n,j) * binomial(Bell.(x)+j, n) umbrally where Bell_n(x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; and Lag(n,x,m), the associated Laguerre polynomials of order m.
%F A008292 For x = 0, the equation gives Sum_{j=0..n} E(n,j) * binomial(j,n) = 1 for n=0 and 0 for all other n. By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*binomial(y,n), for x in the equation, the Worpitzky identity is obtained; y^n = Sum_{j=0..n} E(n,j) * binomial(y+j,n).
%F A008292 Note that E(n,j)/n! = E(n,j)/(Sum_{k=0..n} E(n,k)). Also (n!*Lag(n, -1, j-n)) is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to the equation for x=1; n!*Bell_n(1) = n!*Sum_{j=0..n} S2(n,j) = Sum_{j=0..n} E(n,j) * (n!*Lag(n, -1, j-n)).
%F A008292 (Appended Sep 16 2020) For connections to the Bernoulli numbers, extensions, proofs, and a clear presentation of the number arrays involved in the identities above, see my post Reciprocity and Umbral Witchcraft. (End)
%F A008292 From the relations between the h- and f-polynomials of permutohedra and reciprocals of e.g.f.s described in A049019: (t-1)((t-1)d/dx)^n 1/(t-exp(x)) evaluated at x=0 gives the n-th Eulerian row polynomial in t and the n-th row polynomial in (t-1) of A019538 and A090582. From the Comtet and Copeland references in A139605: ((t+exp(x)-1)d/dx)^(n+1) x gives pairs of the Eulerian polynomials in t as the coefficients of x^0 and x^1 in its Taylor series expansion in x. - _Tom Copeland_, Oct 05 2008
%F A008292 G.f: 1/(1-x/(1-x*y/1-2*x/(1-2*x*y/(1-3*x/(1-3*x*y/(1-... (continued fraction). - _Paul Barry_, Mar 24 2010
%F A008292 If n is odd prime, then the following consecutive 2*n+1 terms are 1 modulo n: a((n-1)*(n-2)/2+i), i=0..2*n. This chain of terms is maximal in the sense that neither the previous term nor the following one are 1 modulo n. - _Vladimir Shevelev, Jul 01 2011
%F A008292 From _Peter Bala_, Sep 29 2011: (Start)
%F A008292 For k = 0,1,2,... put G(k,x,t) := x -(1+2^k*t)*x^2/2 +(1+2^k*t+3^k*t^2)*x^3/3-(1+2^k*t+3^k*t^2+4^k*t^3)*x^4/4+.... Then the series reversion of G(k,x,t) with respect to x gives an e.g.f. for the present table when k = 0 and for A008517 when k = 1.
%F A008292 The e.g.f. B(x,t) := compositional inverse with respect to x of G(0,x,t) = (exp(x)-exp(x*t))/(exp(x*t)-t*exp(x)) = x + (1+t)*x^2/2! + (1+4*t+t^2)*x^3/3! + ... satisfies the autonomous differential equation dB/dx = (1+B)*(1+t*B) = 1 + (1+t)*B + t*B^2.
%F A008292 Applying [Bergeron et al., Theorem 1] gives a combinatorial interpretation for the Eulerian polynomials: A(n,t) counts plane increasing trees on n vertices where each vertex has outdegree <= 2, the vertices of outdegree 1 come in 1+t colors and the vertices of outdegree 2 come in t colors. An example is given below. Cf. A008517. Applying [Dominici, Theorem 4.1] gives the following method for calculating the Eulerian polynomials: Let f(x,t) = (1+x)*(1+t*x) and let D be the operator f(x,t)*d/dx. Then A(n+1,t) = D^n(f(x,t)) evaluated at x = 0.
%F A008292 (End)
%F A008292 With e.g.f. A(x,t) = G[x,(t-1)]-1 in Copeland's 2008 comment, the compositional inverse is Ainv(x,t) = log(t-(t-1)/(1+x))/(t-1). - _Tom Copeland_, Oct 11 2011
%F A008292 T(2*n+1,n+1) = (2*n+2)*T(2*n,n). (E.g., 66 = 6*11, 2416 = 8*302, ...) - _Gary Detlefs_, Nov 11 2011
%F A008292 E.g.f.: (1-y) / (1 - y*exp( (1-y)*x )). - _Geoffrey Critzer_, Nov 10 2012
%F A008292 From _Peter Bala_, Mar 12 2013: (Start)
%F A008292 Let {A(n,x)} n>=1 denote the sequence of Eulerian polynomials beginning [1, 1 + x, 1 + 4*x + x^2, ...]. Given two complex numbers a and b, the polynomial sequence defined by R(n,x) := (x+b)^n*A(n+1,(x+a)/(x+b)), n >= 0, satisfies the recurrence equation R(n+1,x) = d/dx((x+a)*(x+b)*R(n,x)). These polynomials give the row generating polynomials for several triangles in the database including A019538 (a = 0, b = 1), A156992 (a = 1, b = 1), A185421 (a = (1+i)/2, b = (1-i)/2), A185423 (a = exp(i*Pi/3), b = exp(-i*Pi/3)) and A185896 (a = i, b = -i).
%F A008292 (End)
%F A008292 E.g.f.: 1 + x/(T(0) - x*y), where T(k) = 1 + x*(y-1)/(1 + (k+1)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 07 2013
%F A008292 From _Tom Copeland_, Sep 18 2014: (Start)
%F A008292 A) Bivariate e.g.f. A(x,a,b)= (e^(ax)-e^(bx))/(a*e^(bx)-b*e^(ax)) = x + (a+b)*x^2/2! + (a^2+4ab+b^2)*x^3/3! + (a^3+11a^2b+11ab^2+b^3)x^4/4! + ...
%F A008292 B) B(x,a,b)= log((1+ax)/(1+bx))/(a-b) = x - (a+b)x^2/2 + (a^2+ab+b^2)x^3/3 - (a^3+a^2b+ab^2+b^3)x^4/4 + ... = log(1+u.*x), with (u.)^n = u_n = h_(n-1)(a,b) a complete homogeneous polynomial, is the compositional inverse of A(x,a,b) in x (see Drake, p. 56).
%F A008292 C) A(x) satisfies dA/dx = (1+a*A)(1+b*A) and can be written in terms of a Weierstrass elliptic function (see Buchstaber & Bunkova).
%F A008292 D) The bivariate Eulerian row polynomials are generated by the iterated derivative ((1+ax)(1+bx)d/dx)^n x evaluated at x=0 (see A145271).
%F A008292 E) A(x,a,b)= -(e^(-ax)-e^(-bx))/(a*e^(-ax)-b*e^(-bx)), A(x,-1,-1) = x/(1+x), and B(x,-1,-1) = x/(1-x).
%F A008292 F) FGL(x,y) = A(B(x,a,b) + B(y,a,b),a,b) = (x+y+(a+b)xy)/(1-ab*xy) is called the hyperbolic formal group law and related to a generalized cohomology theory by Lenart and Zainoulline. (End)
%F A008292 For x > 1, the n-th Eulerian polynomial A(n,x) = (x - 1)^n * log(x) * Integral_{u>=0} (ceiling(u))^n * x^(-u) du. - _Peter Bala_, Feb 06 2015
%F A008292 Sum_{j>=0} j^n/e^j, for n>=0, equals Sum_{k=1..n} T(n,k)e^k/(e-1)^(n+1), a rational function in the variable "e" which evaluates, approximately, to n! when e = A001113 = 2.71828... - _Richard R. Forberg_, Feb 15 2015
%F A008292 For a fixed k, T(n,k) ~ k^n, proved by induction. - _Ran Pan_, Oct 12 2015
%F A008292 From A145271, multiply the n-th diagonal (with n=0 the main diagonal) of the lower triangular Pascal matrix by g_n = (d/dx)^n (1+a*x)*(1+b*x) evaluated at x= 0, i.e., g_0 = 1, g_1 = (a+b), g_2 = 2ab, and g_n = 0 otherwise, to obtain the tridiagonal matrix VP with VP(n,k) = binomial(n,k) g_(n-k). Then the m-th bivariate row polynomial of this entry is P(m,a,b) = (1, 0, 0, 0, ...) [VP * S]^(m-1) (1, a+b, 2ab, 0, ...)^T, where S is the shift matrix A129185, representing differentiation in the divided powers basis x^n/n!. Also, P(m,a,b) = (1, 0, 0, 0, ...) [VP * S]^m (0, 1, 0, ...)^T. - _Tom Copeland_, Aug 02 2016
%F A008292 Cumulatively summing a row generates the n starting terms of the n-th differences of the n-th powers. Applying the finite difference method to x^n, these terms correspond to those before constant n! in the lowest difference row. E.g., T(4,k) is summed as 0+1=1, 1+11=12, 12+11=23, 23+1=4!. See A101101, A101104, A101100, A179457. - _Andy Nicol_, May 25 2024
%e A008292 The triangle T(n, k) begins:
%e A008292 n\k 1 2 3 4 5 6 7 8 9 10 ...
%e A008292 1: 1
%e A008292 2: 1 1
%e A008292 3: 1 4 1
%e A008292 4: 1 11 11 1
%e A008292 5: 1 26 66 26 1
%e A008292 6: 1 57 302 302 57 1
%e A008292 7: 1 120 1191 2416 1191 120 1
%e A008292 8: 1 247 4293 15619 15619 4293 247 1
%e A008292 9: 1 502 14608 88234 156190 88234 14608 502 1
%e A008292 10: 1 1013 47840 455192 1310354 1310354 455192 47840 1013 1
%e A008292 ... Reformatted. - _Wolfdieter Lang_, Feb 14 2015
%e A008292 -----------------------------------------------------------------
%e A008292 E.g.f. = (y) * x^1 / 1! + (y + y^2) * x^2 / 2! + (y + 4*y^2 + y^3) * x^3 / 3! + ... - _Michael Somos_, Mar 17 2011
%e A008292 Let n=7. Then the following 2*7+1=15 consecutive terms are 1(mod 7): a(15+i), i=0..14. - _Vladimir Shevelev_, Jul 01 2011
%e A008292 Row 3: The plane increasing 0-1-2 trees on 3 vertices (with the number of colored vertices shown to the right of a vertex) are
%e A008292 .
%e A008292 . 1o (1+t) 1o t 1o t
%e A008292 . | / \ / \
%e A008292 . | / \ / \
%e A008292 . 2o (1+t) 2o 3o 3o 2o
%e A008292 . |
%e A008292 . |
%e A008292 . 3o
%e A008292 .
%e A008292 The total number of trees is (1+t)^2 + t + t = 1 + 4*t + t^2.
%p A008292 A008292 := proc(n,k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1,k)+(n-k+1)*procname(n-1,k-1) ; end if; end proc:
%t A008292 t[n_, k_] = Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j, 0, k}];
%t A008292 Flatten[Table[t[n, k], {n, 1, 10}, {k, 1, n}]] (* _Jean-François Alcover_, May 31 2011, after _Michael Somos_ *)
%t A008292 Flatten[Table[CoefficientList[(1-x)^(k+1)*PolyLog[-k, x]/x, x], {k, 1, 10}]] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%t A008292 Table[Tally[
%t A008292 Count[#, x_ /; x > 0] & /@ (Differences /@
%t A008292 Permutations[Range[n]])][[;; , 2]], {n, 10}] (* _Li Han_, Oct 11 2020 *)
%o A008292 (PARI) {T(n, k) = if( k<1 || k>n, 0, if( n==1, 1, k * T(n-1, k) + (n-k+1) * T(n-1, k-1)))}; /* _Michael Somos_, Jul 19 1999 */
%o A008292 (PARI) {T(n, k) = sum( j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))}; /* _Michael Somos_, Jul 19 1999 */
%o A008292 (PARI) {A(n,c)=c^(n+c-1)+sum(i=1,c-1,(-1)^i/i!*(c-i)^(n+c-1)*prod(j=1,i,n+c+1-j))}
%o A008292 (Haskell)
%o A008292 import Data.List (genericLength)
%o A008292 a008292 n k = a008292_tabl !! (n-1) !! (k-1)
%o A008292 a008292_row n = a008292_tabl !! (n-1)
%o A008292 a008292_tabl = iterate f [1] where
%o A008292 f xs = zipWith (+)
%o A008292 (zipWith (*) ([0] ++ xs) (reverse ks)) (zipWith (*) (xs ++ [0]) ks)
%o A008292 where ks = [1 .. 1 + genericLength xs]
%o A008292 -- _Reinhard Zumkeller_, May 07 2013
%o A008292 (Python)
%o A008292 from sympy import binomial
%o A008292 def T(n, k): return sum([(-1)**j*(k - j)**n*binomial(n + 1, j) for j in range(k + 1)])
%o A008292 for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Nov 08 2017
%o A008292 (R)
%o A008292 T <- function(n, k) {
%o A008292 S <- numeric()
%o A008292 for (j in 0:k) S <- c(S, (-1)^j*(k-j)^n*choose(n+1, j))
%o A008292 return(sum(S))
%o A008292 }
%o A008292 for (n in 1:10){
%o A008292 for (k in 1:n) print(T(n,k))
%o A008292 } # _Indranil Ghosh_, Nov 08 2017
%o A008292 (GAP) Flat(List([1..10],n->List([1..n],k->Sum([0..k],j->(-1)^j*(k-j)^n*Binomial(n+1,j))))); # _Muniru A Asiru_, Jun 29 2018
%o A008292 (Sage) [[sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k)) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Feb 23 2019
%o A008292 (Magma) Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >; [[Eulerian(n,k): k in [0..n-1]]: n in [1..10]]; // _G. C. Greubel_, Apr 15 2019
%Y A008292 Columns k=2..8 are A000295, A000460, A000498, A000505, A000514, A001243, A001244.
%Y A008292 Cf. A019538, A028246, A048993, A048994, A049019, A086885, A090582, A129185, A131758, A139605, A173018.
%K A008292 nonn,tabl,nice,eigen,core,look
%O A008292 1,5
%A A008292 _N. J. A. Sloane_, Mar 15 1996
%E A008292 Thanks to _Michael Somos_ for additional comments.
%E A008292 Further comments from _Christian G. Bower_, May 12 2000
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