A223901 Poly-Cauchy numbers of the second kind hat c_n^(-3).

1, -8, 35, -161, 854, -5248, 36966, -294714, 2628600, -25963392, 281529192, -3326287848, 42546905712, -585889457328, 8643254959008, -136013600978784, 2274436197944064, -40278639752011008, 753115809287568384, -14826614346669090816, 306574242780102220800

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).
• Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Crossrefs

Cf. A081046, A223899, A223902, A223904.

Cf. A222636.

Programs

Magma

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^3: k in [0..n]]: n in [0..25]];

Mathematica

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

PARI

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

Formula

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

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

From Seiichi Manyama, Apr 15 2025: (Start)

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

a(n) = (-1)^n * Sum_{k=0..3} k! * Stirling2(4,k+1) * |Stirling1(n+1,k+1)|. (End)