A006233 Denominators of Cauchy numbers of first type.

1, 2, 6, 4, 30, 4, 84, 24, 90, 20, 132, 8, 5460, 840, 360, 48, 1530, 4, 1596, 168, 1980, 1320, 8280, 80, 81900, 6552, 1512, 112, 3480, 80, 114576, 7392, 117810, 7140, 1260, 8, 3838380, 5928, 936, 48, 81180, 440, 1191960, 55440, 869400, 38640, 236880, 224

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• nice
• easy

Comments

The corresponding numerators are given in A006232.

The signed rationals A006232(n)/a(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.

Examples

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

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
• H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
• L. Jolley, Summation of Series, Chapman and Hall, London, 1925, pp. 14-15 (formula 70).
• 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..1000
• A. Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind, J. Number Theory, 73 (1998), 47-58.
• I. S. Gradsteyna and I. M. Ryzhik, Table of integrals, series and products, (1980), page 2 (formula 0.131).
• Wolfdieter Lang, Sheffer a- and z-sequences
• 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. 306 (2006), no. 16, 1906-1920.
• Eric Weisstein's World of Mathematics, Bernoulli Number of the Second Kind
• Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., 45 (2007), 146-150.
• Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., 309 (2009), 3830-3842.

Crossrefs

Cf. A006232 (numerators), A002206, A002207, A002208, A002209, A002657, A002790.

Programs

Magma

[Denominator((&+[StirlingFirst(n,k)/(k+1): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Nov 13 2018

Maple

seq(denom(add(stirling1(n,k)/(k+1),k=0..n)),n=0..12); # Peter Luschny, Apr 28 2009

Mathematica

With[{nn=50},Denominator[CoefficientList[Series[x/Log[1+x],{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Oct 28 2011 *)
a[n_] := Sum[ StirlingS1[n, k]/(k+1), {k, 0, n}] // Denominator; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 10 2013, after Peter Luschny *)
Join[{1}, Array[Abs@Denominator[ Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 50]] (* Michael De Vlieger, Nov 13 2018 *)

PARI

for(n=0,50, print1(denominator( 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
print(0, 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,den//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 += 1 # A.H.M. Smeets, Nov 14 2018

Python

from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006233(n): return sum(Fraction(stirling(n,k,kind=1,signed=True),k+1) for k in range(n+1)).denominator # Chai Wah Wu, Jul 09 2023

Sage

def A006233_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).denominator())
        f *= n+1
    return R
print(A006233_list(50))  # G. C. Greubel, Nov 13 2018

Formula

Denominator of integral of x(x-1)...(x-n+1) from 0 to 1.

E.g.f.: x/log(1+x).

Denominator of Sum_{k=0..n} A048994(n,k)/(k+1). [Peter Luschny, Apr 28 2009]

a(n) = denominator(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/(2*n) - Sum_{k >= 2} abs((A006232(k)/a(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018