A096341 E.g.f.: exp(x)/(1-x)^7.

1, 8, 71, 694, 7421, 86276, 1084483, 14665106, 212385209, 3280842496, 53862855551, 936722974958, 17205245113141, 332864226563324, 6766480571358971, 144202473398010826, 3215159679583864433

Offset: 0

Author

Philippe Deléham, Jun 28 2004

Keywords

• nonn

Comments

Sum_{k=0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n) for x = 1, 2, 3, 4, 5, 6 respectively.

Links

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

Crossrefs

Cf. E.g.f. exp(x)/(1-x)^k: A000522 (k = 1), A001339 (k = 2), A082030 (k = 3), A095000 (k = 4), A095177 (k = 5), A096307 (k = 6).

Cf. A094795, A094816.

Programs

Mathematica

Table[HypergeometricPFQ[{7, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
With[{nn = 250}, CoefficientList[Series[Exp[x]/(1 - x)^7, {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 27 2016 *)

Formula

a(n) = Sum_{k = 0..n} A094816(n, k)*7^k.

a(n) = Sum_{k = 0..n} binomial(n, k)*(k+6)!/6!.

a(n) = 2F0(7,-n;;-1). - Benedict W. J. Irwin, May 27 2016

From Peter Bala, Jul 26 2021: (Start)

a(n) = (n+7)*a(n-1) - (n-1)*a(n-2) with a(0) = 1 and a(1) = 8.

First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) + 1 with a(0) = 1, where P(n) = n^6 + 15*n^5 + 100*n^4 + 355*n^3 + 694*n^2 + 689*n + 265 = A094795(n).

(End)