A113296 Cumulative product of double factorial A006882.

1, 1, 2, 6, 48, 720, 34560, 3628800, 1393459200, 1316818944000, 5056584744960000, 52563198423859200000, 2422112183371431936000000, 327312129899898454671360000000, 211155601241022491077587763200000000

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

Cf. A006882, A074962, A137988, A137989.

Sequence in context: A129464 A175430 A003053 * A275462 A063744 A141609

Adjacent sequences: A113293 A113294 A113295 * A113297 A113298 A113299

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 18 2006

Status

approved