A222636 Poly-Cauchy numbers c_n^(-3).

1, 8, 19, -1, -10, 48, -234, 1302, -8328, 60672, -497688, 4547448, -45846864, 505862064, -6065584128, 78555965184, -1093053332736, 16264215348480, -257730606190080, 4333624828853760, -77067187081620480, 1445257352902763520, -28505367984508416000

Offset: 0

Author

Takao Komatsu, Mar 28 2013

Keywords

• sign

Comments

Definition of poly-Cauchy numbers in A222627.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012)
• Takao 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.
• Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
• M. Z. Spivey,Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146.
• Wikipedia, Stirling transform

Crossrefs

Column k=3 of A383049.

Cf. A223901.

Programs

Magma

[&+[StirlingFirst(n,k)*(k+1)^3: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

Mathematica

Table[Sum[StirlingS1[n, k] (k + 1)^3, {k, 0, n}], {n, 0, 25}]

PARI

a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^3); \\ Michel Marcus, Nov 14 2015

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^3.

E.g.f.: (1 + x) * (1 + 7 * log(1 + x) + 6 * log(1 + x)^2 + log(1 + x)^3). - Ilya Gutkovskiy, Aug 10 2021

E.g.f.: Sum_{k>=0} (k+1)^3 * log(1+x)^k / k!. - Seiichi Manyama, Apr 14 2025