Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Oct 21 2021 01:25:47
%S 0,1,10,78,568,4120,30864,244720,2088832,19389312,196514560,
%T 2173194496,26128665600,339890756608,4759410116608,71395178280960,
%U 1142340032364544,19419853564641280,349557673401188352,6641597100292636672,132831947503410872320,2789470920661372502016
%N a(n) = n! * Sum_{k=0..n-1} 4^k / k!.
%F E.g.f.: x * exp(4*x) / (1 - x).
%F a(0) = 0; a(n) = n * (a(n-1) + 4^(n-1)).
%F a(n) ~ exp(4)*n!. - _Stefano Spezia_, Oct 11 2021
%t Table[n! Sum[4^k/k!, {k, 0, n - 1}], {n, 0, 21}]
%t nmax = 21; CoefficientList[Series[x Exp[4 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) a(n) = n!*sum(k=0, n-1, 4^k/k!); \\ _Michel Marcus_, Oct 11 2021
%Y Cf. A007526, A053487, A066534, A348312.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Oct 11 2021