login
Hankel transform of A186001.
2

%I #22 Nov 22 2023 21:33:05

%S 1,1,2,16,768,294912,1132462080,52183852646400,33664847019245568000,

%T 347485857744891213250560000,64560982045934655213753964953600000,

%U 239901585047846581083822477336190648320000000

%N Hankel transform of A186001.

%H G. C. Greubel, <a href="/A186002/b186002.txt">Table of n, a(n) for n = 0..39</a>

%F a(n) = Product_{k=0..n} (2*k+0^k)^(n-k).

%F a(n+1) = 2^C(n+1,2)*Product_(k!,k,1,n) = A000178(n)*A006125(n+1).

%F Essentially the same as A108400.

%F From _Alexander R. Povolotsky_, Feb 10 2011: (Start)

%F WolframAlpha shows that

%F a(n) = (0^n*2^(1/2*(n-1)*n)*exp^(1/12-zeta^(1, 0)(-1, n+1)))/A

%F where zeta(s, a)is the generalized Riemann zeta function and A is the Glaisher‐Kinkelin constant.

%F WolframAlpha suggests that for all terms given

%F a(n) = 2^(1/2*(n-1)*n)*G(n+1)

%F where G(n) is the Barnes G-function. (End)

%F 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 Table[2^(1/2*(n - 1)*n)*BarnesG[n + 1], {n, 0, 25}] (* _G. C. Greubel_, Feb 22 2017 *)

%Y Cf. A000178, A006125, A074962, A108400, A186001.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Feb 09 2011