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%I #6 Jan 23 2024 08:40:41
%S 1,10,7000,882000000,37784880000000000,890287342560000000000000000,
%T 16991329795972963200000000000000000000000,
%U 363197259318543010730772480000000000000000000000000000000
%N Hankel transform of quintuple factorial numbers A047055.
%F 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)}.
%F 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
%t Table[Product[(Floor[5*(2*k+1)/2]*5*(k+1))^(n-k), {k,0,n}], {n,0,10}] (* _Vaclav Kotesovec_, Jan 23 2024 *)
%Y Cf. A047055.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Dec 03 2009
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