A201367 E.g.f.: 3*exp(3*x) / (5 - 2*exp(3*x)).

1, 5, 35, 345, 4515, 73905, 1451835, 33273945, 871529715, 25681042305, 840815302635, 30281769805545, 1189735610250915, 50638609760802705, 2321120945531697435, 113992686944812385145, 5971520591679167948115, 332369999588147180115105, 19587647624733292373756235

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..250

Formula

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+3*k*x).

O.g.f.: A(x) = 1/(1 - 5*x/(1-2*x/(1 - 10*x/(1-4*x/(1 - 15*x/(1-6*x/(1 - 20*x/(1-8*x/(1 - 25*x/(1-10*x/(1 - ...))))))))))), a continued fraction.

a(n) = Sum_{k=0..n} (-3)^(n-k) * 5^k * Stirling2(n,k) * k!.

a(n) = Sum_{k=0..n} A123125(n,k)*5^k*2*(n-k). - Philippe Deléham, Nov 30 2011

a(n) ~ n! / (2*(log(5/2)/3)^(n+1)). - Vaclav Kotesovec, Jun 13 2013

a(n) = 3^n*log(5/2) * Integral_{x = 0..oo} (ceiling(x))^n * (5/2)^(-x) dx. - Peter Bala, Feb 06 2015

Example

E.g.f.: E(x) = 1 + 5*x + 35*x^2/2! + 345*x^3/3! + 4515*x^4/4! + 73905*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 35*x^2 + 345*x^3 + 4515*x^4 + 73905*x^5 + ...
where A(x) = 1 + 5*x/(1+3*x) + 2!*5^2*x^2/((1+3*x)*(1+6*x)) + 3!*5^3*x^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*5^4*x^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + ...

Maple

S:= series(3*exp(3*x)/(5-2*exp(3*x)), x, 51):
seq(coeff(S, x, n)*n!, n=0..50); # Robert Israel, Nov 18 2019

Mathematica

Table[Sum[(-3)^(n-k)*5^k*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 13 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(3*exp(3*x+x*O(x^n))/(5 - 2*exp(3*x+x*O(x^n))), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}
(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-3)^(n-k)*5^k*Stirling2(n, k)*k!)}

Crossrefs

Cf. A201365, A201366, A201368.

Sequence in context: A346666 A369724 A360611 * A233860 A258902 A371028

Adjacent sequences: A201364 A201365 A201366 * A201368 A201369 A201370

Keyword

nonn,easy

Author

Paul D. Hanna, Nov 30 2011

Status

approved