A283812 Poly-Bernoulli numbers B_n^(k) with k = -6.

1, 64, 1394, 20266, 237686, 2441314, 22934774, 202229266, 1701740006, 13821281314, 109214866454, 844558486066, 6419351203526, 48118995192514, 356641942834934, 2618939805811666, 19085432672558246, 138206899494338914, 995563711729120214, 7139963278111582066, 51017526215427244166

Offset: 0

Author

Seiichi Manyama, Mar 17 2017

Keywords

• nonn
• easy

Comments

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{6,n}. - Vincent Pilaud, Sep 16 2020

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1179
• Index entries for linear recurrences with constant coefficients, signature (27,-295,1665,-5104,8028,-5040).

Crossrefs

Row 6 of array A099594.

Programs

Mathematica

Table[720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n , {n, 0, 18}] (* Indranil Ghosh, Mar 17 2017 *)

PARI

a(n) = 720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n; \\ Indranil Ghosh, Mar 17 2017

PARI

Vec((1 - x)^2*(1 + 39*x + 38*x^2 - 120*x^3) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^20)) \\ Colin Barker, Oct 14 2020

Python

def A283812(n): return 720*7**n - 1800*6**n + 1560*5**n - 540*4**n + 62*3**n - 2**n # Indranil Ghosh, Mar 17 2017

Formula

a(n) = 720*7^n - 1800*6^n + 1560*5^n - 540*4^n + 62*3^n - 2^n.

From Colin Barker, Oct 14 2020: (Start)

G.f.: (1 - x)^2*(1 + 39*x + 38*x^2 - 120*x^3) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).

a(n) = 27*a(n-1) - 295*a(n-2) + 1665*a(n-3) - 5104*a(n-4) + 8028*a(n-5) - 5040*a(n-6) for n>5. (End)

E.g.f.: exp(2*x)*(720*exp(5*x) - 1800*exp(4*x) + 1560*exp(3*x) - 540*exp(2*x) + 62*exp(x) - 1). - Stefano Spezia, May 18 2024