A223904 Poly-Cauchy numbers of the second kind hat c_n^(-5).

1, -32, 275, -1817, 12134, -87784, 699894, -6158058, 59566464, -630057696, 7246806720, -90151868160, 1207028135520, -17314992935040, 265048030579680, -4313510679824160, 74387763047472000, -1355291635314213120, 26016022725597866880, -524865277479851360640, 11103724030717930095360

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, A223902.

Cf. A223023.

Programs

Magma

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^5: k in [0..n]]: n in [0..23]]; // Vincenzo Librandi, Aug 03 2013

Mathematica

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^5, {k, 0, n}], {n, 0, 30}]

PARI

a(n) = sum(k=0, n, (-1)^k*stirling(n, k, 1)*(k+1)^5); \\ Michel Marcus, Nov 14 2015

Formula

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^5 * Stirling1(n,k).

From Seiichi Manyama, Apr 15 2025: (Start)

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

E.g.f.: (1/(1+x)) * Sum_{k=0..5} Stirling2(6,k+1) * (-log(1+x))^k.

a(n) = (-1)^n * Sum_{k=0..5} k! * Stirling2(6,k+1) * |Stirling1(n+1,k+1)|. (End)