A283813 Poly-Bernoulli numbers B_n^(k) with k = -7.

1, 128, 4246, 85310, 1315666, 17234438, 202229266, 2193664790, 22447207906, 219680806598, 2076319823986, 19088476874870, 171615294959746, 1515094215592358, 13177154171845906, 113190802751806550, 962272631860465186, 8109687887324611718, 67845242760941615026

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_{7,n}. - Vincent Pilaud, Sep 16 2020

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1103
• Index entries for linear recurrences with constant coefficients, signature (35,-511,4025,-18424,48860,-69264,40320).

Crossrefs

Row 7 of array A099594.

Programs

Mathematica

Table[5040*8^n - 15120*7^n + 16800*6^n - 8400*5^n + 1806*4^n - 126*3^n + 2^n , {n, 0, 18}] (* Indranil Ghosh, Mar 17 2017 *)
LinearRecurrence[{35,-511,4025,-18424,48860,-69264,40320},{1,128,4246,85310,1315666,17234438,202229266},30] (* Harvey P. Dale, Oct 29 2020 *)

PARI

a(n) = 5040*8^n - 15120*7^n + 16800*6^n - 8400*5^n + 1806*4^n - 126*3^n + 2^n ; \\ Indranil Ghosh, Mar 17 2017

PARI

Vec((1 - x)*(1 + 94*x + 371*x^2 - 1546*x^3 + 1200*x^4) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)) + O(x^20)) \\ Colin Barker, Oct 14 2020

Python

def A283813(n): return 5040*8**n - 15120*7**n + 16800*6**n - 8400*5**n + 1806*4**n - 126*3**n + 2**n # Indranil Ghosh, Mar 17 2017

Formula

a(n) = 5040*8^n - 15120*7^n + 16800*6^n - 8400*5^n + 1806*4^n - 126*3^n + 2^n.

From Colin Barker, Oct 14 2020: (Start)

G.f.: (1 - x)*(1 + 94*x + 371*x^2 - 1546*x^3 + 1200*x^4) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)).

a(n) = 35*a(n-1) - 511*a(n-2) + 4025*a(n-3) - 18424*a(n-4) + 48860*a(n-5) - 69264*a(n-6) + 40320*a(n-7) for n>6. (End)

E.g.f.: exp(2*x)*(5040*exp(6*x) - 15120*exp(5*x) + 16800*exp(4*x) - 8400*exp(3*x) + 1806*exp(2*x) - 126*exp(x) + 1). - Stefano Spezia, May 18 2024