A203516 - OEIS

%I #25 Feb 19 2024 14:54:00

%S 1,4,192,184320,4954521600,4794391461888000,204135216112950312960000,

%T 451965950843675288237663846400000,

%U 60040562704967329457107799785403842560000000,542366306792798635131534558788357929673196306432000000000

%N a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

%C Each term divides its successor, as in A034910.

%C See A093883 for a guide to related sequences.

%H G. C. Greubel, <a href="/A203516/b203516.txt">Table of n, a(n) for n = 1..33</a>

%F a(n) ~ sqrt(A) * 2^(-7/24 - n + 3*n^2/2) * exp(-1/24 + n/2 - 3*n^2/4) * n^(1/24 - n/2 + n^2/2), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Sep 01 2023

%F From _G. C. Greubel_, Feb 19 2024: (Start)

%F a(n) = BarnesG(n+1)*A203517(n).

%F a(n) = 2^binomial(n,2) * Product_{j=1..n-1} (2j)!/j!. (End)

%p a:= n-> mul(mul(2*(i+j-1), i=1..j-1), j=2..n):

%p seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017

%t f[j_] := 2 j - 1; z = 15;

%t v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]

%t d[n_] := Product[(i - 1)!, {i, 1, n}]    (* A000178 *)

%t Table[v[n], {n, 1, z}]                   (* A203516 *)

%t Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}]  (* A034910 *)

%t Table[v[n]/d[n], {n, 1, 20}]             (* A203517 *)

%t Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n]/E^(1/8), {n, 1, 10}] (* _Vaclav Kotesovec_, Sep 01 2023 *)

%o (PARI) a(n) = my(pd=1); for(j=1, n, for(i=1, j-1, pd=pd*2*(i+j-1))); pd \\ _Felix Fröhlich_, Jul 23 2017

%o (Magma) [2^Binomial(n,2)*(&*[Factorial(2*k)/Factorial(k): k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Feb 19 2024

%o (SageMath) [2^binomial(n,2)*product(factorial(2*k)/factorial(k) for k in range(n)) for n in range(1,21)] # _G. C. Greubel_, Feb 19 2024

%Y Cf. A000178, A005408, A034910, A093883, A203516, A203517.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 03 2012

%E Name edited by _Alois P. Heinz_, Jul 23 2017