A169621 Hankel transform of quintuple factorial numbers A047055.

1, 10, 7000, 882000000, 37784880000000000, 890287342560000000000000000, 16991329795972963200000000000000000000000, 363197259318543010730772480000000000000000000000000000000

Offset: 0

Author

Paul Barry, Dec 03 2009

Keywords

• easy
• nonn

Crossrefs

Cf. A047055.

Programs

Mathematica

Table[Product[(Floor[5*(2*k+1)/2]*5*(k+1))^(n-k), {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Jan 23 2024 *)

Formula

a(n)=Product{k=0..n, (floor(5(2k+1)/2)*floor(5(2k+2)/2))^(n-k)}=Product{k=0..n, (floor(5(2k+1)/2)*5(k+1))^(n-k)}.

a(n) ~ (2*Pi)^(n + 7/10) * 5^(n*(n+1)) * n^(n^2 + 7*n/5 + 31/75) / (A * Gamma(2/5)^(n + 2/5) * exp(3*n^2/2 + 7*n/5 - 1/12 - c)), where A is the Glaisher-Kinkelin constant A074962 and c = zeta'(-1, 2/5) = 0.0827672925828924139907562934385991589097620172389278574723... - Vaclav Kotesovec, Jan 23 2024