A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).

1, -16, 97, -531, 3148, -20940, 156680, -1310840, 12166096, -124281120, 1387313520, -16813355280, 219967479744, -3090914335104, 46439677053120, -743069262651840, 12616998421804416, -226608929801923968, 4292762009479969536, -85545808260446050560, 1789078468694176410624

Offset: 0

Author

Takao Komatsu, Mar 29 2013

Keywords

• sign

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
• Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012), p. 42-53.
• Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Crossrefs

Cf. A081046, A223899, A223901, A223904.

Cf. A222748.

Programs

Mathematica

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^4, {k, 0, n}], {n, 0, 30}]

PARI

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

Formula

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

From Seiichi Manyama, Apr 15 2025: (Start)

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

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

a(n) = (-1)^n * Sum_{k=0..4} k! * Stirling2(5,k+1) * |Stirling1(n+1,k+1)|. (End)