A374574 - OEIS

%I #19 Jul 12 2024 19:11:39

%S 1,3,32,870,46224,4037880,522956160,93928267440,22324392518400,

%T 6780385526302080,2561327494111411200,1177652997443424902400,

%U 647478071469567800985600,419450149241406188889984000,316196664211373618844934963200,274410818470142134209609852672000

%N a(n) = Sum_{j=n..2n} j!.

%H Alois P. Heinz, <a href="/A374574/b374574.txt">Table of n, a(n) for n = 0..224</a>

%F a(n) = a(n-1) - (n-1)! + (2*n-1)! + (2*n)! with a(0) = 1.

%F a(n) = Sum_{j=0..n} (n + j)!.

%F a(n) = A100822(2n,n).

%F a(n) = A143122(2n,n).

%p a:= proc(n) option remember; `if`(n<3, [1, 3, 32][n+1],

%p      ((16*n^3-16*n^2-n+2)*a(n-1)-(n-1)*(16*n^3-20*n^2+6*n-1)

%p       *a(n-2)+2*(2*n-1)*(4*n+1)*(n-1)*(n-2)*a(n-3))/(4*n-3))

%p     end:

%p seq(a(n), n=0..15);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       a(n-1) -(n-1)! +(2*n-1)! +(2*n)!)

%p     end:

%p seq(a(n), n=0..15);

%Y Row sums of A143084.

%Y Cf. A000142, A100822, A143122, A296591 (the same for product).

%Y Diagonal of A054115, A211370.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Jul 11 2024