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A057863 a(n) = Product_{k=1..n} (2k-1)!!. 16
1, 1, 3, 45, 4725, 4465125, 46414974375, 6272287562165625, 12714083695698776015625, 438120013555654794702228515625, 286849911214281324754704976473779296875, 3943988517696329309474874414036059896739501953125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the coefficient of the closed form for BarnesG[(2n-1)/2].

a(n) is the hook product corresponding to the partition (n,n-1,...,2,1). a(n)=(n(n+1)/2)!/A005118(n+1). - Emeric Deutsch, May 21 2004

Hankel transform of A185998. - Paul Barry, Feb 08 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..35

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, Barnes G-Function

FORMULA

a(n) = Product_{k=0..n} (2*k+1)^(n-k).

a(n) ~ A^(1/2) * 2^(n^2/2+n+5/24) * n^(n^2/2+n/2+1/24) / exp(3*n^2/4+n/2+1/24), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

a(n) = 2^(n*(n+1)/2-1/24) * A^(3/2) * Pi^(-n/2-1/4) * exp(-1/8) * G(n+3/2), where A is the Glaisher-Kinkelin constant, G is the Barnes G-function. - Vladimir Reshetnikov, Nov 06 2015

a(n) = sqrt(G(2*n+2)) / (2^(n^2/2) * G(n+1) * sqrt(Gamma(n+1))), where G is the Barnes G-function. - Vaclav Kotesovec, Apr 08 2021

MAPLE

a:= n-> mul((2*k+1)^(n-k), k=0..n):

seq(a(n), n=0..15);  # Alois P. Heinz, Nov 28 2012

MATHEMATICA

a[n_] := Product[2^i Gamma[1/2+i]/Sqrt[Pi], {i, n}]

Table[Product[(2*k+1)^(n-k), {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 13 2014 *)

Table[Product[(2k-1)!!, {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 13 2014 *)

Table[2^(n(n+1)/2-1/24) Glaisher^(3/2) Pi^(-n/2-1/4) E^(-1/8) BarnesG[n+3/2], {n, 0, 10}] (* Vladimir Reshetnikov, Nov 06 2015 *)

Table[Sqrt[BarnesG[2*n + 2]] / (2^(n^2/2) * BarnesG[n+1] * Sqrt[Gamma[n+1]]), {n, 0, 12}] (* Vaclav Kotesovec, Apr 08 2021 *)

PROG

(PARI) a(n)=prod(k=0, n-1, prod(i=0, k, 2*i+1))

CROSSREFS

Cf. A000178, A005118, A074962, A089626.

Sequence in context: A004105 A060336 A268196 * A302156 A229415 A265621

Adjacent sequences:  A057860 A057861 A057862 * A057864 A057865 A057866

KEYWORD

nonn,changed

AUTHOR

Eric W. Weisstein

EXTENSIONS

Simpler description from Benoit Cloitre, May 03 2003

Definition and programs corrected by Vaclav Kotesovec, Nov 13 2014

STATUS

approved

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Last modified April 12 12:30 EDT 2021. Contains 342920 sequences. (Running on oeis4.)