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A306644 a(n) = Sum_{k=0..n} (n^2)!/(k! * (n-k)!)^n. 3

%I #18 Jun 21 2021 03:51:58

%S 1,2,36,94080,114144030000,128569399991042250240,

%T 231970526672859167062880173363200,

%U 974076884952864555606703666490413198470021120000,13999785014750877128592398884910508842895938385473568105272652000000

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

%H Seiichi Manyama, <a href="/A306644/b306644.txt">Table of n, a(n) for n = 0..24</a>

%F 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

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

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

%Y Main diagonal of A306641.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 02 2019

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)