A223851 Poly-Cauchy numbers c_4^(-n).

2, 4, -10, -116, -538, -1316, 3110, 68044, 593702, 4117564, 25598630, 149298604, 835538342, 4545417244, 24232768550, 127291207564, 661279993382, 3406640457724, 17437000616870, 88808710210924, 450563930438822, 2278977841239004, 11499752014647590

Offset

1,1

Comments

The poly-Cauchy numbers c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))*(-1)^m/(m+1)^k, m=0..n).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012)

TTakao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.

Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

M. Z. Spivey, Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146

Wikipedia, Stirling transform

Formula

Empirical g.f.: 2*x*(38*x^2-12*x+1) / ((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Mar 31 2013

Mathematica

Table[Sum[StirlingS1[4, k] (k + 1)^n, {k, 0, 4}], {n, 25}]

Prog

(Magma) [&+[StirlingFirst(4, k)*(k+1)^n: k in [0..4]]: n in [1..25]]; // Bruno Berselli, Mar 28 2013
(PARI) a(n) = sum(k=0, 4, stirling(4, k, 1)*(k+1)^n); \\ Michel Marcus, Nov 14 2015

Crossrefs

Sequence in context: A270479 A126140 A326967 * A371621 A297364 A355203

Adjacent sequences: A223848 A223849 A223850 * A223852 A223853 A223854

Keyword

sign

Author

Takao Komatsu, Mar 28 2013

Status

approved