A202950 a(n) = Sum_{k=0..n} (2*n-k)!*2^(k-n)/k!.

1, 2, 10, 127, 3251, 138826, 8853202, 786297485, 92660657077, 13979292051826, 2626450694785226, 601179186815081227, 164665579315045664935, 53172608709697779630602, 19988633342014049108880226, 8653593506915464727302042201, 4274276964973547062653005905577, 2389044486666800863650341729928610, 1500174879893101746801192365463624202

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..531

Formula

Appears to satisfy recurrence

a(n+3) = (12 + 11 n + 2 n^2) a(n+2) + (5 n + 2 n^2) a(n+1) - a(n) + 8

corresponding to differential equation for g.f.

(5*t^2-2*t^3-3*t^4)*(d/dt)a(t)-2*t^3*(-1+t^2)*(d^2/dt^2)a(t)+(-1-t-t^2+2*t^3+t^4)*a(t)+1+3*t+3*t^2+t^3.

Apparently also a(n) + (-2*n^2+n+2)*a(n-1) + (2*n-3)*a(n-2) +(2*n^2-11*n+13)*a(n-3) - a(n-4) = 0. - R. J. Mathar, May 19 2014

a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n + 1/2) / exp(2*n). - Vaclav Kotesovec, Dec 20 2017

Example

a(2) = 4!/(4*0!) + 3!/(2*1!) + 2!/(1*2!) = 10.

Maple

A:= n -> add((2*n-k)!*2^(k-n)/k!, k=0..n)

Mathematica

Table[Sum[(n+k)! / (2^k * (n-k)!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 20 2017 *)

Prog

(PARI) a(n)=sum(k=0, n, (2*n-k)!<<(k-n)/k!) \\ Charles R Greathouse IV, Jan 03 2012

Crossrefs

Sequence in context: A333455 A372316 A334555 * A144835 A305028 A119191

Adjacent sequences: A202947 A202948 A202949 * A202951 A202952 A202953

Keyword

nonn,easy

Author

Robert Israel, Dec 30 2011

Status

approved