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%I A145905 #12 Feb 04 2022 06:56:56
%S A145905 1,1,1,1,3,1,1,9,5,1,1,27,25,7,1,1,81,125,49,9,1,1,243,625,343,81,11,
%T A145905 1,1,729,3125,2401,729,121,13,1,1,2187,15625,16807,6561,1331,169,15,1,
%U A145905 1,6561,78125,117649,59049,14641,2197,225,17,1,1,19683,390625,823543
%N A145905 Square array read by antidiagonals: Hilbert transform of triangle A060187.
%C A145905 Definition of the Hilbert transform of a triangular array:
%C A145905 For many square arrays in the database the entries in a row are polynomial in the column index, of degree d say and hence the row generating function has the form P(x)/(1-x)^(d+1), where P is some polynomial function. Often the array whose rows are formed from the coefficients of these P polynomials is of independent interest. This suggests the following definition.
%C A145905 Let [L(n,k)]n,k>=0 be a lower triangular array and let R(n,x) := sum {k = 0 .. n} L(n,k)*x^k, denote the n-th row generating polynomial of L. Then we define the Hilbert transform of L, denoted Hilb(L), to be the square array whose n-th row, n >= 0, has the generating function R(n,x)/(1-x)^(n+1).
%C A145905 In this particular case, L is the array A060187, the array of Eulerian numbers of type B, whose row polynomials are the h-polynomials for permutohedra of type B. The Hilbert transform is an infinite Vandermonde matrix V(1,3,5,...).
%C A145905 We illustrate the Hilbert transform with a few examples:
%C A145905 (1) The Delannoy number array A008288 is the Hilbert transform of Pascal's triangle A007318 (view as the array of coefficients of h-polynomials of n-dimensional cross polytopes).
%C A145905 (2) The transpose of the array of nexus numbers A047969 is the Hilbert transform of the triangle of Eulerian numbers A008292 (best viewed in this context as the coefficients of h-polynomials of n-dimensional permutohedra of type A).
%C A145905 (3) The sequence of Eulerian polynomials begins [1, x, x + x^2, x + 4*x^2 + x^3, ...]. The coefficients of these polynomials are recorded in triangle A123125, whose Hilbert transform is A004248 read as square array.
%C A145905 (4) A108625, the array of crystal ball sequences for the A_n lattices, is the Hilbert transform of A008459 (viewed as the triangle of coefficients of h-polynomials of n-dimensional associahedra of type B).
%C A145905 (5) A142992, the array of crystal ball sequences for the C_n lattices, is the Hilbert transform of A086645, the array of h-vectors for type C root polytopes.
%C A145905 (6) A108553, the array of crystal ball sequences for the D_n lattices, is the Hilbert transform of A108558, the array of h-vectors for type D root polytopes.
%C A145905 (7) A086764, read as a square array, is the Hilbert transform of the rencontres numbers A008290.
%C A145905 (8) A143409 is the Hilbert transform of triangle A073107.
%H A145905 Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H A145905 S. Parker, <a href="http://people.brandeis.edu/~gessel/homepage/students/parkerthesis.pdf">The Combinatorics of Functional Composition and Inversion</a>, Ph.D. Dissertation, Brandeis Univ. (1993) [From _Tom Copeland_, Nov 09 2008]
%F A145905 T(n,k) = (2*k + 1)^n, (see equation 4.10 in [Franssens]). This array is the infinite Vandermonde matrix V(1,3,5,7, ....) having a LDU factorization equal to A039755 * diag(2^n*n!) * transpose(A007318).
%e A145905 Triangle A060187 (with an offset of 0) begins
%e A145905 1;
%e A145905 1, 1;
%e A145905 1, 6, 1;
%e A145905 so the entries in the first three rows of the Hilbert transform of
%e A145905 A060187 come from the expansions:
%e A145905 Row 0: 1/(1-x) = 1 + x + x^2 + x^3 + ...;
%e A145905 Row 1: (1+x)/(1-x)^2 = 1 + 3*x + 5*x^2 + 7*x^3 + ...;
%e A145905 Row 2: (1+6*x+x^2)/(1-x)^3 = 1 + 9*x + 25*x^2 + 49*x^3 + ...;
%e A145905 The array begins
%e A145905 n\k|..0....1.....2.....3......4
%e A145905 ================================
%e A145905 0..|..1....1.....1.....1......1
%e A145905 1..|..1....3.....5.....7......9
%e A145905 2..|..1....9....25....49.....81
%e A145905 3..|..1...27...125...343....729
%e A145905 4..|..1...81...625..2401...6561
%e A145905 5..|..1..243..3125.16807..59049
%e A145905 ...
%p A145905 T:=(n,k) -> (2*k + 1)^n: seq(seq(T(n-k,k),k = 0..n),n = 0..10);
%Y A145905 Cf. A008292, A039755, A052750 (first superdiagonal), A060187, A114172, A145901.
%K A145905 easy,nonn,tabl
%O A145905 0,5
%A A145905 _Peter Bala_, Oct 27 2008

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