A096307 E.g.f.: exp(x)/(1-x)^6.

1, 7, 55, 481, 4645, 49171, 566827, 7073725, 95064361, 1369375615, 21054430591, 344231563897, 5964569413645, 109196040092491, 2106381399472435, 42705264827626261, 907920105215691217, 20198878182718877815

Offset: 0

Author

Philippe Deléham, Jun 26 2004

Keywords

• nonn
• easy

Comments

Sum_{k = 0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n), A095000(n), A095177(n) for x = 1, 2, 3, 4, 5 respectively.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A000522, A001339, A082030, A095000, A095177, A001689, A094794.

Programs

Mathematica

Table[HypergeometricPFQ[{6, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
With[{nn = 250}, CoefficientList[Series[Exp[x]/(1 - x)^6, {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 27 2016 *)

Formula

a(n) = Sum_{k = 0..n} A094916(n, k)*6^k.

a(n) = Sum_{k = 0..n} binomial(n, k)*(k+5)!/5!.

a(n) = 2F0(6,-n;;-1). - Benedict W. J. Irwin, May 27 2016

From Peter Bala, Jul 25 2021: (Start)

a(n) = (n+6)*a(n-1) - (n-1)*a(n-2) with a(0) = 1 and a(1) = 7. Cf. A001689.

First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) - 1 with a(0) = 1, where P(n) = n^5 + 10*n^4 + 45*n^3 + 100*n^2 + 109*n + 44 = A094794(n).

(End)