cp's OEIS Frontend

A186002 Hankel transform of A186001.

%I A186002 #25 Apr 27 2025 08:03:08
%S A186002 1,1,2,16,768,294912,1132462080,52183852646400,33664847019245568000,
%T A186002 347485857744891213250560000,64560982045934655213753964953600000,
%U A186002 239901585047846581083822477336190648320000000
%N A186002 Hankel transform of A186001.
%H A186002 G. C. Greubel, <a href="/A186002/b186002.txt">Table of n, a(n) for n = 0..39</a>
%F A186002 a(n) = Product_{k=0..n} (2*k+0^k)^(n-k).
%F A186002 a(n+1) = 2^C(n+1,2)*Product_(k!,k,1,n) = A000178(n)*A006125(n+1).
%F A186002 Essentially the same as A108400.
%F A186002 From _Alexander R. Povolotsky_, Feb 10 2011: (Start)
%F A186002 WolframAlpha shows that
%F A186002 a(n) = (0^n*2^(1/2*(n-1)*n)*exp^(1/12-zeta^(1, 0)(-1, n+1)))/A
%F A186002 where zeta(s, a) is the generalized Riemann zeta function and A is the Glaisher-Kinkelin constant.
%F A186002 WolframAlpha suggests that for all terms given
%F A186002 a(n) = 2^(1/2*(n-1)*n)*G(n+1)
%F A186002 where G(n) is the Barnes G-function. (End)
%F A186002 a(n) ~ 2^(n^2/2) * n^(n^2/2 - 1/12) * Pi^(n/2) / (A * exp(3*n^2/4 - 1/12)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Feb 24 2019
%t A186002 Table[2^(1/2*(n - 1)*n)*BarnesG[n + 1], {n, 0, 25}] (* _G. C. Greubel_, Feb 22 2017 *)
%Y A186002 Cf. A000178, A006125, A074962, A108400, A186001.
%K A186002 nonn,easy
%O A186002 0,3
%A A186002 _Paul Barry_, Feb 09 2011