%I #44 Dec 20 2017 05:10:00
%S 1,2,10,127,3251,138826,8853202,786297485,92660657077,13979292051826,
%T 2626450694785226,601179186815081227,164665579315045664935,
%U 53172608709697779630602,19988633342014049108880226,8653593506915464727302042201,4274276964973547062653005905577,2389044486666800863650341729928610,1500174879893101746801192365463624202
%N a(n) = Sum_{k=0..n} (2*n-k)!*2^(k-n)/k!.
%H Robert Israel, <a href="/A202950/b202950.txt">Table of n, a(n) for n = 0..531</a>
%F Appears to satisfy recurrence
%F a(n+3) = (12 + 11 n + 2 n^2) a(n+2) + (5 n + 2 n^2) a(n+1) - a(n) + 8
%F corresponding to differential equation for g.f.
%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.
%F 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
%F a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n + 1/2) / exp(2*n). - _Vaclav Kotesovec_, Dec 20 2017
%e a(2) = 4!/(4*0!) + 3!/(2*1!) + 2!/(1*2!) = 10.
%p A:= n -> add((2*n-k)!*2^(k-n)/k!,k=0..n)
%t Table[Sum[(n+k)! / (2^k * (n-k)!), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 20 2017 *)
%o (PARI) a(n)=sum(k=0,n,(2*n-k)!<<(k-n)/k!) \\ _Charles R Greathouse IV_, Jan 03 2012
%K nonn,easy
%O 0,2
%A _Robert Israel_, Dec 30 2011