A222627 Poly-Cauchy numbers c_n^(-2) (for definition see Comments lines).

1, 4, 5, -3, 4, -8, 20, -52, 72, 936, -17568, 238752, -3113280, 41503680, -577877760, 8470414080, -131039838720, 2139954163200, -36854615347200, 668374040678400, -12742107588403200, 254904791591116800, -5341386032640000000, 117034910701793280000

Offset: 0

Author

Takao Komatsu, Mar 28 2013

Keywords

• sign

Comments

The definition of poly-Cauchy numbers is given in Theorem 1 of the paper Poly-Cauchy numbers (see Links lines).

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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=2 of A383049.

Cf. A006233.

Programs

Magma

[&+[StirlingFirst(n,k)*(k+1)^2: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

Mathematica

Table[Sum[StirlingS1[n, k]*(k + 1)^2, {k, 0, n}], {n, 0, 25}]

PARI

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

Formula

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

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

E.g.f.: Sum_{k>=0} (k+1)^2 * log(1+x)^k / k!. - Seiichi Manyama, Apr 14 2025