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
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).
Crossrefs
Programs
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Magma
[24*5^n-36*4^n+14*3^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011
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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);
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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 *) -
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
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SageMath
[24*5^n -36*4^n +14*3^n -2^n for n in (0..30)] # G. C. Greubel, Aug 02 2022
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
Comments