A001908 E.g.f. exp(-x)/(1-5*x).

1, 4, 41, 614, 12281, 307024, 9210721, 322375234, 12895009361, 580275421244, 29013771062201, 1595757408421054, 95745444505263241, 6223453892842110664, 435641772498947746481, 32673132937421080986074, 2613850634993686478885921, 222177303974463350705303284

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100

Crossrefs

Column k=5 of A320032.

Programs

Maple

f:= gfun:-rectoproc({a(n+1) = (5*n+4)* a(n) + 5*n*a(n-1),a(0)=1,a(1)=4},a(n),remember):
seq(f(n),n=0..30); # Robert Israel, Aug 06 2015

Mathematica

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[-x]/(1 - 5 x), {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)

PARI

x='x+O('x^33); Vec(serlaplace(exp(-x)/(1-5*x))) \\ Gheorghe Coserea, Aug 06 2015

Formula

E.g.f.: A(x) = exp(-x)/(1-5x) satisfies (1-5x)A' - (4+5x)A = 0. - Gheorghe Coserea, Aug 06 2015

a(n+1) = (5n+4) a(n) + 5n a(n-1). - Gheorghe Coserea, Aug 06 2015

a(n) = 5^n*exp(-1/5)*Gamma(n+1,-1/5), where Gamma is the incomplete Gamma function. - Robert Israel, Aug 06 2015

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020