A223173 Poly-Cauchy numbers c_3^(-n).

-1, -3, -1, 45, 359, 2037, 10079, 46365, 204119, 873477, 3666959, 15191085, 62342279, 254119317, 1030760639, 4165958205, 16792710839, 67557739557, 271392171119, 1089053371725, 4366669645799, 17498051254197, 70086331418399, 280627721655645

Offset: 1

Author

Takao Komatsu, Mar 28 2013

Keywords

• sign
• changed

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
• Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Programs

Magma

[&+[StirlingFirst(3,k)*(k+1)^n: k in [0..3]]: n in [1..25]]; // Bruno Berselli, Mar 28 2013

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}]

PARI

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

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