A186002 Hankel transform of A186001.

1, 1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000

Offset: 0

Author

Paul Barry, Feb 09 2011

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..39

Crossrefs

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

Programs

Mathematica

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

Formula

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

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

Essentially the same as A108400.

From Alexander R. Povolotsky, Feb 10 2011: (Start)

WolframAlpha shows that

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

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

WolframAlpha suggests that for all terms given

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

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

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