A223852 Poly-Cauchy numbers c_5^(-n).

-6, -8, 48, 340, 984, -1148, -34152, -254780, -1250376, -3417788, 12508248, 296104900, 3122953464, 26485493572, 201873508248, 1443404093380, 9892106472504, 65798800964932, 428187502981848, 2740792716574660, 17321987718906744, 108394003491348292

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, 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.
• 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 (20,-155,580,-1044,720).

Programs

Magma

[&+[StirlingFirst(5,k)*(k+1)^n: k in [0..5]]: n in [1..25]]; // Bruno Berselli, Mar 28 2013

Mathematica

Table[Sum[StirlingS1[5, k] (k + 1)^n, {k, 0, 5}], {n, 25}]

PARI

a(n) = sum(k=0, 5, stirling(5, k, 1)*(k+1)^n); \\ Michel Marcus, Nov 14 2015

Formula

a(n) = Sum_{k=0..5} Stirling1(5,k)*(k+1)^n.

Empirical g.f.: -2*x*(810*x^3 - 361*x^2 + 56*x - 3) / ((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Mar 31 2013