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 *)
With[{nn=20}, CoefficientList[Series[(3*Exp[3x])/(5-2*Exp[3x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Sep 07 2024 *)
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
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Nov 30 2011
STATUS
approved