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A211374 Product of all the parts in the partitions of n into exactly 2 parts.

%I A211374 #42 Mar 10 2022 03:17:35
%S A211374 1,1,2,12,24,360,720,20160,40320,1814400,3628800,239500800,479001600,
%T A211374 43589145600,87178291200,10461394944000,20922789888000,
%U A211374 3201186852864000,6402373705728000,1216451004088320000,2432902008176640000,562000363888803840000
%N A211374 Product of all the parts in the partitions of n into exactly 2 parts.
%H A211374 G. C. Greubel, <a href="/A211374/b211374.txt">Table of n, a(n) for n = 1..449</a>
%H A211374 <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%F A211374 a(n) = ( (n-1)! * floor(n/2)! )/( n-1-floor(n/2) )!.
%F A211374 a(n) = P(n-1, floor(n/2)) * floor(n/2)!, where P(n,k) are the k-permutations of n objects. - _Wesley Ivan Hurt_, Jun 07 2013
%F A211374 a(2n) = A002674(n); a(2n+1) = A010050(n). - _Wesley Ivan Hurt_, Oct 16 2014
%F A211374 a(n) = Product_{i=1..floor(n/2)} i * (n-i). - _Wesley Ivan Hurt_, Nov 14 2017
%F A211374 From _Amiram Eldar_, Mar 10 2022: (Start)
%F A211374 Sum_{n>=1} 1/a(n) = 3*cosh(1) - 2.
%F A211374 Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - cosh(1). (End)
%e A211374 Define a(1):=1; a(2) = 1 since 2 = 1+1 and (1)*(1) = 1; a(3) = 2 since 3 = 2+1 and (2)*(1) = 2; a(4) = 12 since 4 = 3+1 = 2+2 and (3)*(1)*(2)*(2) = 12; a(5) = 24 since 5 = 4+1 = 3+2 and (4)*(1)*(3)*(2) = 24.
%p A211374 A211374:=n->( (n-1)! * floor(n/2)! )/( (n-1) - floor(n/2) )!: seq(A211374(k), k=1..25);
%p A211374 with(combinat, numbperm): seq(numbperm(k-1, floor(k/2))*floor(k/2)!, k = 1..25); # _Wesley Ivan Hurt_, Jun 07 2013
%t A211374 Table[Times @@ Flatten[Select[Partitions[n], Length[#] == 2 &]], {n, 25}] (* _T. D. Noe_, Feb 11 2013 *)
%t A211374 Table[((n - 1)!*Floor[n/2]!)/(n - 1 - Floor[n/2])!, {n, 25}] (* _Wesley Ivan Hurt_, Oct 16 2014 *)
%o A211374 (Magma) [(Factorial(n-1) * Factorial(Floor(n/2)))/Factorial(n-1-Floor(n/2)) : n in [1..25]]; // _Wesley Ivan Hurt_, Oct 16 2014
%o A211374 (PARI) a(n) = prod(i=1, n\2, i*(n-i)); \\ _Michel Marcus_, Nov 14 2017
%Y A211374 Cf. A001318, A002620, A081123, A081125, A093353.
%Y A211374 Bisections: A002674, A010050.
%K A211374 nonn,easy
%O A211374 1,3
%A A211374 _Wesley Ivan Hurt_, Feb 06 2013