login
E.g.f. exp(-x)/(1-5*x).
(Formerly M3677 N1500)
6

%I M3677 N1500 #35 Jan 17 2020 10:36:50

%S 1,4,41,614,12281,307024,9210721,322375234,12895009361,580275421244,

%T 29013771062201,1595757408421054,95745444505263241,

%U 6223453892842110664,435641772498947746481,32673132937421080986074,2613850634993686478885921,222177303974463350705303284

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

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001908/b001908.txt">Table of n, a(n) for n = 0..100</a>

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

%F a(n+1) = (5n+4) a(n) + 5n a(n-1). - _Gheorghe Coserea_, Aug 06 2015

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

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (5*k - 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020

%p 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):

%p seq(f(n),n=0..30); # _Robert Israel_, Aug 06 2015

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

%o (PARI) x='x+O('x^33); Vec(serlaplace(exp(-x)/(1-5*x))) \\ _Gheorghe Coserea_, Aug 06 2015

%Y Column k=5 of A320032.

%K nonn

%O 0,2

%A _N. J. A. Sloane_