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

1, 1, 2, 12, 24, 360, 720, 20160, 40320, 1814400, 3628800, 239500800, 479001600, 43589145600, 87178291200, 10461394944000, 20922789888000, 3201186852864000, 6402373705728000, 1216451004088320000, 2432902008176640000, 562000363888803840000

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..449

Index entries for sequences related to partitions.

Formula

a(n) = ( (n-1)! * floor(n/2)! )/( n-1-floor(n/2) )!.

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

a(2n) = A002674(n); a(2n+1) = A010050(n). - Wesley Ivan Hurt, Oct 16 2014

a(n) = Product_{i=1..floor(n/2)} i * (n-i). - Wesley Ivan Hurt, Nov 14 2017

From Amiram Eldar, Mar 10 2022: (Start)

Sum_{n>=1} 1/a(n) = 3*cosh(1) - 2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - cosh(1). (End)

Example

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.

Maple

A211374:=n->( (n-1)! * floor(n/2)! )/( (n-1) - floor(n/2) )!: seq(A211374(k), k=1..25);
with(combinat, numbperm): seq(numbperm(k-1, floor(k/2))*floor(k/2)!, k = 1..25); # Wesley Ivan Hurt, Jun 07 2013

Mathematica

Table[Times @@ Flatten[Select[Partitions[n], Length[#] == 2 &]], {n, 25}] (* T. D. Noe, Feb 11 2013 *)
Table[((n - 1)!*Floor[n/2]!)/(n - 1 - Floor[n/2])!, {n, 25}] (* Wesley Ivan Hurt, Oct 16 2014 *)

Prog

(Magma) [(Factorial(n-1) * Factorial(Floor(n/2)))/Factorial(n-1-Floor(n/2)) : n in [1..25]]; // Wesley Ivan Hurt, Oct 16 2014
(PARI) a(n) = prod(i=1, n\2, i*(n-i)); \\ Michel Marcus, Nov 14 2017

Crossrefs

Cf. A001318, A002620, A081123, A081125, A093353.

Bisections: A002674, A010050.

Sequence in context: A052565 A176710 A141900 * A126962 A002207 A181814

Adjacent sequences: A211371 A211372 A211373 * A211375 A211376 A211377

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 06 2013

Status

approved