A202879 E.g.f.: exp(4*x + x^2/2).

1, 4, 17, 76, 355, 1724, 8671, 45028, 240809, 1323460, 7461121, 43079084, 254388667, 1534503676, 9445067375, 59263320964, 378729294481, 2463130313348, 16290919259569, 109500022678540, 747527556645971, 5180110680154684, 36418521410184127, 259636520604139556

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = Sum_{k=0..[n/2]} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!).

O.g.f.: 1/(1-4*x - x^2/(1-4*x - 2*x^2/(1-4*x - 3*x^2/(1-4*x - 4*x^2/(1-4*x -...))))), a continued fraction.

Recurrence: a(n) = 4*a(n-1) + (n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ exp(4*sqrt(n)-n/2-4)*n^(n/2)/sqrt(2)*(1+11/(3*sqrt(n))). - Vaclav Kotesovec, Oct 20 2012

Example

E.g.f.: A(x) = 1 + 4*x + 17*x^2/2! + 76*x^3/3! + 355*x^4/4! + 1724*x^5/5! +...

Mathematica

CoefficientList[Series[E^(4*x+x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

Prog

(PARI) {a(n)=n!*polcoeff(exp(4*x+x^2/2+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n\2, 4^(n-2*k)/2^k * n!/((n-2*k)!*k!))}
(PARI) /* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+4*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-4*x-(n-k)*x^2*CF)); polcoeff(CF, n)}

Crossrefs

Cf. A202878.

Sequence in context: A081186 A239204 A005572 * A333059 A081922 A124325

Adjacent sequences: A202876 A202877 A202878 * A202880 A202881 A202882

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Status

approved