A006232 Numerators of Cauchy numbers of first type.

1, 1, -1, 1, -19, 9, -863, 1375, -33953, 57281, -3250433, 1891755, -13695779093, 24466579093, -132282840127, 240208245823, -111956703448001, 4573423873125, -30342376302478019, 56310194579604163

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• frac
• nice

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

Examples

			1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90, ...
		

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.

Crossrefs

Cf. A006233 (denominators), A002206, A002207, A002208, A002209, A002657, A002790.

Programs

Magma

[Numerator((&+[StirlingFirst(n,k)/(k+1): k in [0..n]])): n in [0..20]]; // G. C. Greubel, Nov 13 2018

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 *)

PARI

for(n=0,20, print1(numerator( sum(k=0,n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ 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

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

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