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

0,2

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

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

Mathematica

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

Prog

(Magma) [&+[StirlingFirst(n, k)*(k+1)^2: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013
(PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^2); \\ Michel Marcus, Nov 14 2015

Crossrefs

Cf. A006233.

Sequence in context: A212711 A069197 A021692 * A107793 A275275 A376180

Adjacent sequences: A222624 A222625 A222626 * A222628 A222629 A222630

Keyword

sign

Author

Takao Komatsu, Mar 28 2013

Status

approved