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A006232 Numerators of Cauchy numbers of first type.
(Formerly M5067)
97
1, 1, -1, 1, -19, 9, -863, 1375, -33953, 57281, -3250433, 1891755, -13695779093, 24466579093, -132282840127, 240208245823, -111956703448001, 4573423873125, -30342376302478019, 56310194579604163 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

-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.

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.

H. 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

A. Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind, J. Number Theory, 73 (1998), 47-58.

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.

H.-M. Liu, S-H. Qi, S.-Y. Ding, Some Recurrence Relations for Cauchy Numbers of the First Kind, JIS 13 (2010) # 10.3.8.

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 Numbers 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.

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

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 *)

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

CROSSREFS

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

Sequence in context: A033339 A175674 A175675 * A260328 A122549 A039942

Adjacent sequences:  A006229 A006230 A006231 * A006233 A006234 A006235

KEYWORD

sign,frac,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 20 15:20 EDT 2018. Contains 304339 sequences. (Running on oeis4.)