A057863 a(n) = Product_{k=1..n} (2k-1)!!.

1, 1, 3, 45, 4725, 4465125, 46414974375, 6272287562165625, 12714083695698776015625, 438120013555654794702228515625, 286849911214281324754704976473779296875, 3943988517696329309474874414036059896739501953125

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy

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

The Burchnall-Chaundy polynomials P_n(z) have leading term z^(n^2+n)/a(n). - Michael Somos, Jan 18 2023

Examples

			G.f. = 1 + x + 3*x^2 + 45*x^3 + 4725*x^4 + 4465125*x^5 + ... - _Michael Somos_, Jan 18 2023
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..35
• Alejandro H. Morales, Igor Pak, and Greta Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.
• A. P. Veselov and R. Wilcox, Burchnall-Chaundy polynomials and the Laurent Phenomenon, arXiv:1407.7394 [math-ph], 2014.
• Eric Weisstein's World of Mathematics, Barnes G-Function

Crossrefs

Cf. A000178, A005118, A074962, A086205, A089626.

Programs

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 *)

PARI

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

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

From Michael Somos, Jan 18 2023: (Start)

a(n) = (-1)^floor((n+1)/2)*a(-1-n) for all n in Z.

a(n+1)*a(n-1) = (2*n+1)*a(n)^2 for all n in Z.

(4*n + 8)*a(n+1)^2*a(n+2)^2 = a(n)*a(n+2)^3 + a(n+1)^3*a(n+3) for all n in Z.(End)

a(n) = (1/2^(n*(n-1)/2)) * A086205(n). - Peter Bala, Feb 20 2023

Extensions

Simpler description from Benoit Cloitre, May 03 2003

Definition and programs corrected by Vaclav Kotesovec, Nov 13 2014