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
The Burchnall-Chaundy polynomials P_n(z) have leading term z^(n^2+n)/a(n). - 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
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
EXAMPLE
G.f. = 1 + x + 3*x^2 + 45*x^3 + 4725*x^4 + 4465125*x^5 + ... - Michael Somos, Jan 18 2023
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Simpler description from Benoit Cloitre, May 03 2003
Definition and programs corrected by Vaclav Kotesovec, Nov 13 2014
STATUS
approved