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a(n) = Product_{k=1..n} binomial(2*k,k).
(Formerly M2047)
21

%I M2047 #45 Oct 25 2024 14:27:49

%S 1,2,12,240,16800,4233600,3911846400,13425456844800,

%T 172785629592576000,8400837310791045120000,

%U 1552105098192510332190720000,1094904603628138948657963991040000,2960792853328653706847125274154762240000,30794022150329995743434211126374020153344000000

%N a(n) = Product_{k=1..n} binomial(2*k,k).

%D H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A007685/b007685.txt">Table of n, a(n) for n = 0..50</a>

%H H. W. Gould, <a href="/A007680/a007680.pdf">A class of binomial sums and a series transform</a>, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)

%H Bernd C. Kellner, <a href="https://doi.org/10.5281/zenodo.12167556">Asymptotic products of binomial and multinomial coefficients revisited</a>, Integers 24 (2024), Article #A59, 10 pp.; arXiv:<a href="https://arxiv.org/abs/2312.11369">2312.11369</a> [math.CO], 2023.

%F a(0) = 1, a(n) = (2^(2*n)*a(n - 1)*Gamma(n + 1/2))/(sqrt(Pi)*Gamma(n + 1)). - _Ilya Gutkovskiy_, Sep 18 2015

%F a(n) = (2^(n^2 + n - 1/24)*A^(3/2)*Pi^(-n/2 - 1/4)*BarnesG(n + 3/2))/(e^(1/8)*BarnesG(n + 2)), where A is the Glaisher-Kinkelin constant (A074962), BarnesG is the Barnes G-function. - _Ilya Gutkovskiy_, Sep 18 2015

%F a(n) ~ A^(3/2) * 2^(n^2 + n - 7/24) * exp(n/2 - 1/8) / (Pi^((n+1)/2) * n^(n/2 + 3/8)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Apr 16 2016

%F For n>0, a(n) = 2^((n+1)/2) * sqrt(BarnesG(2*n)) * Gamma(2*n) / (n * BarnesG(n)^2 * Gamma(n)^(7/2)). - _Vaclav Kotesovec_, Apr 20 2024

%F Product_{1 <= j <= i <= n} (i + j)/(i - j + 1). - _Peter Bala_, Oct 25 2024

%p [seq(mul(binomial(2*k,k),k=1..n),n=0..16)];

%t Table[Product[Binomial[2*k, k], {k, 1, n}], {n, 0, 50}] (* _G. C. Greubel_, Feb 02 2017 *)

%o (PARI) a(n) = prod(k=1,n, binomial(2*k, k)); \\ _Michel Marcus_, Sep 18 2015

%Y Cf. A000984, A001142, A112332, A203471, A268196, A296589, A296590, A338550.

%K nonn,easy,changed

%O 0,2

%A _N. J. A. Sloane_