A202832 E.g.f: exp(2*x + 5*x^2/2).

1, 2, 9, 38, 211, 1182, 7639, 50738, 368841, 2767202, 22132249, 182624598, 1582522891, 14122521662, 131109031239, 1250794578818, 12334766500561, 124733099306562, 1297921351160809, 13821821639912198, 150946171640101251, 1684074507271422302, 19217497036753475959

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

Column k=5 of A376826.

Cf. A202831.

Sequence in context: A249925 A370397 A162972 * A069724 A132961 A278050

Adjacent sequences: A202829 A202830 A202831 * A202833 A202834 A202835

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Status

approved