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%I #35 Jul 18 2013 02:35:09
%S 1,2,4,12,54,312,2136,16800,149160,1475280,16081920,191530080,
%T 2473999920,34446303360,514240110720,8193624284160,138780284791680,
%U 2489891543596800,47169750454848000,940914453958617600,19712190644360121600
%N Partial sums of A013999.
%H Vincenzo Librandi, <a href="/A227037/b227037.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = sum(A013999(k), k=0..n).
%F a(n) = sum(sum(C(j-k+1,k)*(-1)^k*(j-k+1)!, k=0..floor((j+1)/2)), j=0..n).
%F Recurrence: a(n+4) -(n+8)*a(n+3) +(3*n+16)*a(n+2) -(3*n+13)*a(n+1) +(n+4)*a(n) = 0.
%F G.f.: Sum_{k>=0} (k+1)!*(x-x^2)^k.
%F a(n) = (n+3)*a(n-1)-2*(n+1)*a(n-2)+(n+1)*a(n-3) for n>2, a(n) = 2^n for n<=2. - _Alois P. Heinz_, Jul 01 2013
%F a(n) ~ n!*n/exp(1). - _Vaclav Kotesovec_, Jul 06 2013
%p a:= proc(n) option remember; `if`(n<3, 2^n,
%p (n+3)*a(n-1) -2*(n+1)*a(n-2) +(n+1)*a(n-3))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 01 2013
%t Table[Sum[Sum[Binomial[j-k+1,k]*(-1)^k*(j-k+1)!,{k,0,Floor[(j+1)/2]}],{j,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 06 2013 *)
%o (Maxima) makelist(sum(sum(binomial(j-k+1,k)*(-1)^k*(j-k+1)!, k,0,floor((j+1)/2)), j, 0, n), n, 0, 20);
%Y Cf. A013999.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Jul 01 2013
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