OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!).
O.g.f.: 1/(1-2*x - 5*x^2/(1-2*x - 10*x^2/(1-2*x - 15*x^2/(1-2*x - 20*x^2/(1-2*x -...))))), a continued fraction.
Recurrence: a(n) = 2*a(n-1) + 5*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ exp(2/5*sqrt(5*n)-n/2-1/5)*5^(n/2)*n^(n/2)/sqrt(2)*(1+17/150*sqrt(5)/sqrt(n)). - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
E.g.f.: 1 + 2*x + 9*x^2/2! + 38*x^3/3! + 211*x^4/4! + 1182*x^5/5! +...
MATHEMATICA
CoefficientList[Series[E^(2*x+5*x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) {a(n)=n!*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n\2, 2^(n-3*k)*5^k*n!/((n-2*k)!*k!))}
(PARI) /* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+2*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-2*x-5*(n-k)*x^2*CF)); polcoeff(CF, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 25 2011
STATUS
approved