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A226731
a(n) = (2n - 1)!/(2n).
1
20, 630, 36288, 3326400, 444787200, 81729648000, 19760412672000, 6082255020441600, 2322315553259520000, 1077167364120207360000, 596585001666576384000000, 388888194657798291456000000
OFFSET
3,1
COMMENTS
For n < 3, the formula does not produce an integer.
The ratio of the product of the partition parts of 2n into exactly two parts to the sum of the partition parts of 2n into exactly two parts. For example, a(3) = 20, and 2*3 = 6 has 3 partitions into exactly two parts: (5,1), (4,2), (3,3). Forming the ratio of product to sum (of parts), we have (5*1*4*2*3*3)/(5+1+4+2+3+3) = 360/18 = 20. - Wesley Ivan Hurt, Jun 24 2013
FORMULA
a(n) = A009445(n-1)/A005843(n) = A002674(n)/A001105(n). - Wesley Ivan Hurt, Jun 24 2013
a(n) ~ sqrt(Pi)*2^(2*n-1)*n^(2*n-3/2)/exp(2*n). - Ilya Gutkovskiy, Nov 01 2016
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=3} 1/a(n) = e - 8/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = cos(1) + sin(1) - 4/3. (End)
EXAMPLE
a(3) = (2*3 - 1)!/(2*3) = 5!/6 = 120/6 = 20.
MAPLE
seq((2*k-1)!/(2*k), k=1..20); # Wesley Ivan Hurt, Jun 24 2013
MATHEMATICA
Table[(2n-1)!/(2n), {n, 3, 20}] (* Harvey P. Dale, Jun 19 2013 *)
PROG
(PARI) a(n) = (2*n-1)!/(2*n); \\ Michel Marcus, Nov 01 2016
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jun 15 2013
STATUS
approved