A027651 Poly-Bernoulli numbers B_n^(k) with k=-4.

1, 16, 146, 1066, 6902, 41506, 237686, 1315666, 7107302, 37712866, 197451926, 1023358066, 5262831302, 26903268226, 136887643766, 693968021266, 3508093140902, 17693879415586, 89084256837206, 447884338361266, 2249284754708102, 11285908565322946, 56587579617416246

Offset

0,2

Comments

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{4,n}. - Vincent Pilaud, Sep 16 2020

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5.

Ken Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.

Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.

Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.

Hiroyuki Komaki, Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index, arXiv:1503.04933 [math.NT], 2015.

Index entries for sequences related to Bernoulli numbers.

Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120).

Formula

a(n) = 24*5^n -36*4^n +14*3^n -2^n. - Vladeta Jovovic, Nov 14 2003

G.f.: (1+4*x)*(1-x)^2/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).

E.g.f.: 24*exp(5*x) - 36*exp(4*x) + 14*exp(3*x) - exp(2*x). - G. C. Greubel, Feb 07 2018

Maple

a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n, j)/(j+1)^k, j=0..n);
seq(a(n, -4), n=0..30);

Mathematica

Table[24*5^n -36*4^n +14*3^n -2^n, {n, 0, 30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{14, -71, 154, -120}, {1, 16, 146, 1066}, 30] (* Harvey P. Dale, Nov 20 2019 *)

Prog

(Magma) [24*5^n-36*4^n+14*3^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011
(PARI) Vec((1+4*x)*((1-x)^2)/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015
(SageMath) [24*5^n -36*4^n +14*3^n -2^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

Crossrefs

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027650.

Row n=4 of array A099594.

Sequence in context: A076029 A196572 A052388 * A125379 A232063 A126537

Adjacent sequences: A027648 A027649 A027650 * A027652 A027653 A027654

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved