OFFSET
0,3
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..57
Alejandro H. Morales, Igor Pak and Greta Panova, Hook formulas for skew shapes III. Multivariate and product formulas, Algebraic Combinatorics, Vol. 2, No. 5 (2019), pp. 815-861; arXiv preprint, arXiv:1707.00931 [math.CO], 2017-2020.
Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.
FORMULA
a(n) = Product_{k=0..n} k!!.
a(n) = n!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
a(n) = n*(n-2)!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
a(n) = 2^((6*n^2+12*n+2-3*(-1)^n)/24) * Pi^(((-1)^n-2*n-3)/8) * exp(-1/8) * A^(3/2) * G((2n+7+(-1)^n)/4) * G((2n+7-(-1)^n)/4), where A is the Glaisher-Kinkelin constant (A074962), G(x) is the Barnes G-function. - Vladimir Reshetnikov, Nov 11 2015
Sum_{n>=0} 1/a(n) = 1/A137989. - Amiram Eldar, Nov 09 2020
Sum_{n>=0} (-1)^n/a(n) = A137988. - Amiram Eldar, Apr 12 2021
EXAMPLE
a(10) = 1!! * 2!! * 3!! * 4!! * 5!! * 6!! * 7!! * 8!! * 9!! * 10!!
= 1 * 2 * 3 * 8 * 15 * 48 * 105 * 384 * 945 * 3840
= 5056584744960000 = 2^23 x 3^9 x 5^4 x 7^2.
MATHEMATICA
Table[Product[k!!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Table[2^((6n^2+12n+2-3(-1)^n)/24) Pi^(((-1)^n-2n-3)/8) Exp[-1/8] Glaisher^(3/2) BarnesG[(2n+7+(-1)^n)/4] BarnesG[(2n+7-(-1)^n)/4], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 11 2015 *)
FoldList[Times, Range[0, 20]!!] (* Harvey P. Dale, Oct 29 2019 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 18 2006
STATUS
approved