A137704 Hankel transform of aerated factorial numbers.

1, 1, 1, 2, 8, 96, 3456, 497664, 286654464, 825564856320, 11888133931008000, 1027134771639091200000, 532466665617704878080000000, 1932215036193527461576704000000000

Offset: 0

Author

Paul Barry, Feb 07 2008

Keywords

• easy
• nonn

Comments

Hankel transform of A084261. Hankel transform of A000142 (n!) with interpolated zeros.

a(n+1) is the Hankel transform of A003319 aerated. [From Paul Barry, Oct 07 2008]

Formula

a(n):=Product{k=0..n, floor((k+2)/2)^(n-k)};

a(n) ~ n^(n^2/2 + 3*n/2 + 7/6) * Pi^(n + 3/2) / (A^4 * 2^(n^2/2 + n/2 - 1/3) * exp(3*n^2/4 + 3*n/2 - 1/3)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jan 20 2024