A223906 Poly-Cauchy numbers of the second kind -hat c_3^(-n).

17, 51, 161, 531, 1817, 6411, 23201, 85731, 322217, 1227771, 4729841, 18379731, 71908217, 282817131, 1116854081, 4424238531, 17567551817, 69882262491, 278365739921, 1109974078131, 4429431765017, 17686337611851, 70651190491361, 282322298874531

Offset: 1

Author

Takao Komatsu, Mar 29 2013

Keywords

• nonn
• easy
• changed

Comments

The poly-Cauchy numbers of the second kind hat c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
• Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012), p. 42-53.
• Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.
• Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Crossrefs

Cf. A223173.

Programs

Mathematica

Table[-Sum[StirlingS1[3, k] (-1)^k (k + 1)^n, {k, 0, 3}], {n, 30}]

PARI

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

Formula

Conjecture: a(n) = 2^(1+n)+3^(1+n)+4^n. G.f.: -x*(144*x^2-102*x+17) / ((2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Mar 31 2013