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
KEYWORD
sign
AUTHOR
Takao Komatsu, Mar 28 2013
STATUS
approved