A365269 a(n) = Product_{k=1..n} A002720(k).

1, 2, 14, 476, 99484, 153802264, 2049722772328, 268353804798726416, 386893462638663037013264, 6798536031341327693983294520096, 1595359632648441879172205168815801694176, 5432770180592069558569584672506997142250856260032

Offset: 0

Author

Vaclav Kotesovec, Aug 30 2023

Keywords

• nonn

Crossrefs

Cf. A002720, A289897.

Programs

Mathematica

Table[Product[k! * LaguerreL[k,-1], {k,1,n}], {n,0,15}]
Table[BarnesG[n+2] * Product[LaguerreL[k,-1], {k,1,n}], {n,0,15}]

Python

from math import prod, factorial, comb
def A365269(n): return prod(sum(factorial(m)*comb(k,m)**2 for m in range(k+1)) for k in range(1,n+1)) # Chai Wah Wu, Aug 31 2023

Formula

log(a(n)) ~ log(BarnesG(n+2)) + 4*n^(3/2)/3 - n*log(n)/4 - (1/4 + log(2) + log(Pi)/2)*n + 55*sqrt(n)/24.

log(a(n)) ~ n^2*log(n)/2 - 3*n^2/4 + 4*n^(3/2)/3 + 3*n*log(n)/4 - (5/4 + log(2)/2)*n + 55*sqrt(n)/24.