OFFSET
0,5
COMMENTS
The corresponding denominators are given in A006233.
-a(n+1), n>=0, also numerators from e.g.f. 1/x-1/log(1+x), with denominators A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/log(1-x) with denominators A075178(n). For formula of unsigned a(n) see A075178.
The signed rationals a(n)/A006233(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences.
Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
Named after the French mathematician, engineer and physicist Augustin-Louis Cauchy (1789-1857). - Amiram Eldar, Jun 17 2021
REFERENCES
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
Harold Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n=0..100
Arnold Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind, J. Number Theory, Vol. 73, No. 1 (1998), pp. 47-58.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 2 (formula 0.131).
L. B. W. Jolley, Summation of Series, Dover, (1961) (formula 70).
Wolfdieter Lang, Sheffer a- and z-sequences.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Hong-Mei Liu, Shu-Hua Qi and Shu-Yan Ding, Some Recurrence Relations for Cauchy Numbers of the First Kind, JIS, Vol. 13 (2010), Article 10.3.8.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
Eric Weisstein's World of Mathematics, Bernoulli Numbers of the Second Kind.
Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., Vol. 45, No. 2 (2007), pp. 146-150.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.
FORMULA
Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x). (Note: the numerator of the coefficient of x^n/n! is a(n) - Michael Somos, Jul 12 2014)
Numerator of Sum_{k=0..n} A048994(n,k)/(k+1). - Peter Luschny, Apr 28 2009
Sum_{k=1..n} 1/k = C + log(n) + 1/(2n) + Sum_{k=2..inf} |a(n)|/A075178(n-1) * 1/(n*(n+1)*...*(n+k-1)) (section 0.131 in Gradshteyn and Ryzhik tables). - Ralf Stephan, Jul 12 2014
a(n) = numerator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Feb 23 2018
Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2n) - Sum_{k >= 2} abs((a(k)/A006233(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018
EXAMPLE
1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90, ...
MAPLE
seq(numer(add(stirling1(n, k)/(k+1), k=0..n)), n=0..20); # Peter Luschny, Apr 28 2009
MATHEMATICA
a[n_] := Numerator[ Sum[ StirlingS1[n, k]/(k + 1), {k, 0, n}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 03 2011, after Maple *)
a[n_] := Numerator[ Integrate[ Gamma[x+1]/Gamma[x-n+1], {x, 0, 1}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 29 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ Integrate[ Pochhammer[ -x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator [ n! SeriesCoefficient[ x / Log[ 1 + x], {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
Join[{1}, Array[Numerator[(1/#) Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 25]] (* Michael De Vlieger, Nov 13 2018 *)
PROG
(Sage)
def A006232_list(len):
f, R, C = 1, [1], [1]+[0]*(len-1)
for n in (1..len-1):
for k in range(n, 0, -1):
C[k] = -C[k-1] * k / (k + 1)
C[0] = -sum(C[k] for k in (1..n))
R.append((C[0]*f).numerator())
f *= n
return R
print(A006232_list(20)) # Peter Luschny, Feb 19 2016
(PARI) for(n=0, 20, print1(numerator( sum(k=0, n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ G. C. Greubel, Nov 13 2018
(Magma) [Numerator((&+[StirlingFirst(n, k)/(k+1): k in [0..n]])): n in [0..20]]; // G. C. Greubel, Nov 13 2018
(Python) # Results are abs values
from fractions import gcd
aa, n, sden = [0, 1], 1, 1
while n < 20:
j, snom, sden, a = 1, 0, (n+1)*sden, 0
while j < len(aa):
snom, j = snom+aa[j]*(sden//(j+1)), j+1
nom, den = snom, sden
print(n, nom//gcd(nom, den))
aa, j = aa+[-aa[j-1]], j-1
while j > 0:
aa[j], j = n*aa[j]-aa[j-1], j-1
n = n+1 # A.H.M. Smeets, Nov 14 2018
(Python)
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006232(n): return sum(Fraction(stirling(n, k, kind=1, signed=True), k+1) for k in range(n+1)).numerator # Chai Wah Wu, Jul 09 2023
CROSSREFS
KEYWORD
sign,frac,nice
AUTHOR
STATUS
approved