|
%I #37 Apr 12 2021 04:38:13
%S 1,1,2,6,48,720,34560,3628800,1393459200,1316818944000,
%T 5056584744960000,52563198423859200000,2422112183371431936000000,
%U 327312129899898454671360000000,211155601241022491077587763200000000
%N Cumulative product of double factorial A006882.
%H Harvey P. Dale, <a href="/A113296/b113296.txt">Table of n, a(n) for n = 0..57</a>
%H Alejandro H. Morales, Igor Pak and Greta Panova, <a href="https://doi.org/10.5802/alco.67">Hook formulas for skew shapes III. Multivariate and product formulas</a>, Algebraic Combinatorics, Vol. 2, No. 5 (2019), pp. 815-861; <a href="https://arxiv.org/abs/1707.00931">arXiv preprint</a>, arXiv:1707.00931 [math.CO], 2017-2020.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.
%F a(n) = Product_{k=0..n} k!!.
%F a(n) = n!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
%F a(n) = n*(n-2)!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
%F 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
%F Sum_{n>=0} 1/a(n) = 1/A137989. - _Amiram Eldar_, Nov 09 2020
%F Sum_{n>=0} (-1)^n/a(n) = A137988. - _Amiram Eldar_, Apr 12 2021
%e a(10) = 1!! * 2!! * 3!! * 4!! * 5!! * 6!! * 7!! * 8!! * 9!! * 10!!
%e = 1 * 2 * 3 * 8 * 15 * 48 * 105 * 384 * 945 * 3840
%e = 5056584744960000 = 2^23 x 3^9 x 5^4 x 7^2.
%t Table[Product[k!!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 17 2015 *)
%t 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 *)
%t FoldList[Times,Range[0,20]!!] (* _Harvey P. Dale_, Oct 29 2019 *)
%Y Cf. A006882, A074962, A137988, A137989.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Feb 18 2006
| 