OFFSET
1,2
COMMENTS
Definition of poly-Cauchy numbers in A222627.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..300
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.
M. Z. Spivey, Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146
Wikipedia, Stirling transform
FORMULA
a(n) = Sum_{k=0..3} Stirling1(3,k)*(k+1)^n.
From Colin Barker, Mar 31 2013: (Start)
Conjecture:
a(n) = 2^(1+n) - 3^(1+n) + 4^n;
g.f.: -x*(6*x-1) / ((2*x-1)*(3*x-1)*(4*x-1)). (End)
Conjecture verified by Robert Israel, Jun 21 2018
MAPLE
seq(2^(n+1)-3^(n+1)+4^n, n=0..30); # Robert Israel, Jun 21 2018
MATHEMATICA
Table[Sum[StirlingS1[3, k] (k + 1)^n, {k, 0, 3}], {n, 25}]
PROG
(Magma) [&+[StirlingFirst(3, k)*(k+1)^n: k in [0..3]]: n in [1..25]]; // Bruno Berselli, Mar 28 2013
(PARI) a(n) = sum(k=0, 3, stirling(3, k, 1)*(k+1)^n); \\ Michel Marcus, Nov 14 2015
CROSSREFS
KEYWORD
sign
AUTHOR
Takao Komatsu, Mar 28 2013
STATUS
approved