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%I #22 Jan 29 2018 06:30:08
%S 0,0,1,2,30,744,403320,482631120,22230943262640,2439304469060699520,
%T 16131709536027319923050880,265557748777251180632423132716800,
%U 382326737887135184960649117960539544556800,1405822033408121123332642294795422193345577766681600
%N a(n) = Sum_{i=1..floor(n/2)} (i*(n-i))!.
%C Sum of the factorials of the products of the parts in each partition of n into two parts.
%H G. C. Greubel, <a href="/A270531/b270531.txt">Table of n, a(n) for n = 0..40</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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
%e 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.
%p A270531:=n->add((i*(n-i))!, i=1..floor(n/2)): seq(A270531(n), n=0..15);
%t Table[Sum[(i*(n - i))!, {i, Floor[n/2]}], {n, 0, 15}]
%o (PARI) a(n) = sum(k=1, n\2, (k*(n-k))!); \\ _Michel Marcus_, Mar 22 2016
%Y Cf. A000142, A023855, A144895.
%K nonn,easy
%O 0,4
%A _Wesley Ivan Hurt_, Mar 18 2016
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