A222748 Poly-Cauchy numbers c_n^(-4).

1, 16, 65, 45, -116, 340, -1240, 5480, -28464, 169248, -1125840, 8197680, -63806016, 514314240, -4058967744, 26952984000, -37203513984, -4251686488704, 140692872720384, -3560137793538048, 84004474130786304, -1955196907518928896, 45927815909901004800

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=4 of A383049.

Programs

Magma

[&+[StirlingFirst(n,k)*(k+1)^4: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

Mathematica

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

PARI

a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^4); \\ Michel Marcus, Nov 14 2015

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^4.

From Seiichi Manyama, Apr 14 2025: (Start)

E.g.f.: Sum_{k>=0} (k+1)^4 * log(1+x)^k / k!.

E.g.f.: (1+x) * Sum_{k=0..4} Stirling2(5,k+1) * log(1+x)^k. (End)