A169619 Hankel transform of quartic (or 4-fold) factorial numbers A007696.

1, 4, 640, 11059200, 39749419008000, 48575443603606732800000, 29918051262318914110928977920000000, 12898757220773940360062849214160935321600000000000

Offset: 0

Author

Paul Barry, Dec 03 2009

Keywords

• easy
• nonn

Crossrefs

Cf. A007696.

Programs

Mathematica

Table[Product[((4*k+1)*(4*k+4))^(n-k), {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Jan 23 2024 *)

Formula

a(n) = Product_{k=0..n} ((4*k+1)*(4*k+4))^(n-k).

a(n) ~ 2^(2*n^2 + 3*n + 5/8) * Pi^(n + 5/8) * n^(n^2 + 5*n/4 + 35/96) / (A^(7/8) * Gamma(1/4)^(n + 1/4) * exp(3*n^2/2 + 5*n/4 - 7/96 - G/(4*Pi))), where G is Catalan's constant A006752 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 23 2024