A227037 Partial sums of A013999.

1, 2, 4, 12, 54, 312, 2136, 16800, 149160, 1475280, 16081920, 191530080, 2473999920, 34446303360, 514240110720, 8193624284160, 138780284791680, 2489891543596800, 47169750454848000, 940914453958617600, 19712190644360121600

Offset: 0

Author

Emanuele Munarini, Jul 01 2013

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A013999.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, 2^n,
      (n+3)*a(n-1) -2*(n+1)*a(n-2) +(n+1)*a(n-3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 01 2013

Mathematica

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 *)

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);

Formula

a(n) = sum(A013999(k), k=0..n).

a(n) = sum(sum(C(j-k+1,k)*(-1)^k*(j-k+1)!, k=0..floor((j+1)/2)), j=0..n).

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.

G.f.: Sum_{k>=0} (k+1)!*(x-x^2)^k.

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

a(n) ~ n!*n/exp(1). - Vaclav Kotesovec, Jul 06 2013