A223899 Poly-Cauchy numbers of the second kind hat c_n^(-2).

1, -4, 13, -51, 244, -1392, 9260, -70508, 605320, -5788008, 61021872, -703384272, 8801449344, -118828732032, 1721888828928, -26656798602240, 439110126743040, -7669109089082880, 141557837068938240, -2753560001544053760, 56299265625742848000

Offset: 0

Author

Takao Komatsu, Mar 29 2013

Keywords

• sign

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, A223901, A223902, A223904.

Cf. A222627.

Programs

Magma

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^2: k in [0..n]]: n in [0..23]]; // Vincenzo Librandi, Aug 03 2013

Mathematica

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^2, {k, 0, n}], {n, 0, 30}]

PARI

a(n) = sum(k=0, n, stirling(n, k, 1)*(-1)^k*(k+1)^2); \\ Michel Marcus, Nov 14 2015

Formula

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^2 * Stirling1(n,k).

E.g.f.: (1 - log(1 + x) * (3 - log(1 + x))) / (1 + x). - Ilya Gutkovskiy, Aug 09 2021

From Seiichi Manyama, Apr 15 2025: (Start)

E.g.f.: Sum_{k>=0} (k+1)^2 * (-log(1+x))^k / k!.

a(n) = (-1)^n * Sum_{k=0..2} k! * Stirling2(3,k+1) * |Stirling1(n+1,k+1)|. (End)