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%I #54 Feb 21 2023 11:31:58
%S 1,1,3,45,4725,4465125,46414974375,6272287562165625,
%T 12714083695698776015625,438120013555654794702228515625,
%U 286849911214281324754704976473779296875,3943988517696329309474874414036059896739501953125
%N a(n) = Product_{k=1..n} (2k-1)!!.
%C a(n) is the coefficient of the closed form for BarnesG[(2n-1)/2].
%C 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
%C Hankel transform of A185998. - _Paul Barry_, Feb 08 2011
%C The Burchnall-Chaundy polynomials P_n(z) have leading term z^(n^2+n)/a(n). - _Michael Somos_, Jan 18 2023
%H Alois P. Heinz, <a href="/A057863/b057863.txt">Table of n, a(n) for n = 0..35</a>
%H Alejandro H. Morales, Igor Pak, and Greta Panova, <a href="https://arxiv.org/abs/1707.00931">Hook formulas for skew shapes III. Multivariate and product formulas</a>, arXiv:1707.00931 [math.CO], 2017.
%H A. P. Veselov and R. Wilcox, <a href="https://arxiv.org/abs/1407.7394">Burchnall-Chaundy polynomials and the Laurent Phenomenon</a>, arXiv:1407.7394 [math-ph], 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>
%F a(n) = Product_{k=0..n} (2*k+1)^(n-k).
%F 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
%F 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
%F 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
%F From _Michael Somos_, Jan 18 2023: (Start)
%F a(n) = (-1)^floor((n+1)/2)*a(-1-n) for all n in Z.
%F a(n+1)*a(n-1) = (2*n+1)*a(n)^2 for all n in Z.
%F (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)
%F a(n) = (1/2^(n*(n-1)/2)) * A086205(n). - _Peter Bala_, Feb 20 2023
%e G.f. = 1 + x + 3*x^2 + 45*x^3 + 4725*x^4 + 4465125*x^5 + ... - _Michael Somos_, Jan 18 2023
%p a:= n-> mul((2*k+1)^(n-k), k=0..n):
%p seq(a(n), n=0..15); # _Alois P. Heinz_, Nov 28 2012
%t a[n_] := Product[2^i Gamma[1/2+i]/Sqrt[Pi], {i, n}]
%t Table[Product[(2*k+1)^(n-k),{k,0,n}],{n,0,10}] (* _Vaclav Kotesovec_, Nov 13 2014 *)
%t Table[Product[(2k-1)!!,{k,1,n}],{n,0,10}] (* _Vaclav Kotesovec_, Nov 13 2014 *)
%t 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 *)
%t 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 *)
%o (PARI) a(n)=prod(k=0,n-1,prod(i=0,k,2*i+1))
%Y Cf. A000178, A005118, A074962, A086205, A089626.
%K nonn,easy
%O 0,3
%A _Eric W. Weisstein_
%E Simpler description from _Benoit Cloitre_, May 03 2003
%E Definition and programs corrected by _Vaclav Kotesovec_, Nov 13 2014
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