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

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).

Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.

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)

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

a(n) = numerator(Sum_{k=0..n} (-1)^k*binomial(2*n,n+k)*Stirling2(n+k,k)/(2-2*n-2*k)), for n>1. - Tani Akinari, Jan 06 2026

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

seq(numer(add(stirling1(n, k)/(k+1), k=0..n)), n=0..20); # Peter Luschny, Apr 28 2009

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 *)

(SageMath)

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

(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

(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

(Maxima) a(n):=if n<2 then 1 else num(sum((-1)^k*binomial(2*n, n+k)*stirling2(n+k, k)/(2-2*n-2*k), k, 0, n)); /* Tani Akinari, Jan 06 2026 */