A270531 a(n) = Sum_{i=1..floor(n/2)} (i*(n-i))!.

0, 0, 1, 2, 30, 744, 403320, 482631120, 22230943262640, 2439304469060699520, 16131709536027319923050880, 265557748777251180632423132716800, 382326737887135184960649117960539544556800, 1405822033408121123332642294795422193345577766681600

Offset: 0

Author

Wesley Ivan Hurt, Mar 18 2016

Keywords

• nonn
• easy

Comments

Sum of the factorials of the products of the parts in each partition of n into two parts.

Examples

			a(4)=30; There are 2 partitions of 4 into two parts: (3,1) and (2,2). The sum of the factorials of the products of the parts in each partition is: (3*1)! + (2*2)! = 3! + 4! = 6 + 24 = 30.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..40
• Index entries for sequences related to partitions

Crossrefs

Cf. A000142, A023855, A144895.

Programs

Maple

A270531:=n->add((i*(n-i))!, i=1..floor(n/2)): seq(A270531(n), n=0..15);

Mathematica

Table[Sum[(i*(n - i))!, {i, Floor[n/2]}], {n, 0, 15}]

PARI

a(n) = sum(k=1, n\2, (k*(n-k))!); \\ Michel Marcus, Mar 22 2016

Formula

a(n) ~ (n^2/4)! ~ sqrt(Pi) * n^(n^2/2+1) / (2^((n^2+1)/2) * exp(n^2/4)) if n is even and a(n) ~ ((n^2-1)/4)! ~ sqrt(Pi) * n^((n^2+1)/2) / (2^(n^2/2) * exp(n^2/4)) if n is odd. - Vaclav Kotesovec, Mar 18 2016