A306644 a(n) = Sum_{k=0..n} (n^2)!/(k! * (n-k)!)^n.

1, 2, 36, 94080, 114144030000, 128569399991042250240, 231970526672859167062880173363200, 974076884952864555606703666490413198470021120000, 13999785014750877128592398884910508842895938385473568105272652000000

Offset: 0

Author

Seiichi Manyama, Mar 02 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..24

Crossrefs

Main diagonal of A306641.

Programs

Mathematica

a[n_] := Sum[(n^2)!/(k! * (n-k)!)^n, {k, 0, n}]; Array[a, 9, 0] (* Amiram Eldar, Jun 21 2021 *)

PARI

{a(n) = sum(k=0, n, (n^2)!/(k!*(n-k)!)^n)}

Formula

a(n) ~ c * 2^(n^2 + 1/2) * n^(n^2 - n + 1) / Pi^(n - 1/2), where c = exp(-1/3)*JacobiTheta3(0, exp(-2)) = exp(-1/3) * EllipticTheta[3, 0, exp(-2)] = 0.910956007080971245990320395256172663671471380838524358269586617628532... if n is even and c = exp(-1/3) * JacobiTheta2(0, exp(-2)) = exp(-1/3) * EllipticTheta[3, 0, exp(-2)] = 0.885121645271745566745223804647879414416684832686710775956467801722557... if n is odd. - Vaclav Kotesovec, Jun 21 2021