login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A113296 Cumulative product of double factorial A006882. 1
1, 1, 2, 6, 48, 720, 34560, 3628800, 1393459200, 1316818944000, 5056584744960000, 52563198423859200000, 2422112183371431936000000, 327312129899898454671360000000, 211155601241022491077587763200000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..57

Alejandro H. Morales, Igor Pak, Greta Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.

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

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.

Sequence in context: A175430 A129464 A003053 * A275462 A063744 A141609

Adjacent sequences:  A113293 A113294 A113295 * A113297 A113298 A113299

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Feb 18 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 17:51 EDT 2020. Contains 333103 sequences. (Running on oeis4.)