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 A306642 a(n) = Sum_{k=0..n} (3*n)!/(k! * (n-k)!)^3. 3

%I #22 Dec 02 2021 13:31:26

%S 1,12,900,94080,11988900,1704214512,260453217024,41886697881600,

%T 6996546610936740,1203384096358158000,211855235800656848400,

%U 38011289046678107596800,6928290032159649797280000,1279703438754969901486464000,239070018975087493229806080000

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

%H Seiichi Manyama, <a href="/A306642/b306642.txt">Table of n, a(n) for n = 0..431</a>

%H Robert W. Donley Jr, <a href="https://arxiv.org/abs/2107.09463">Directed path enumeration for semi-magic squares of size three</a>, arXiv:2107.09463 [math.CO], 2021.

%F a(n) ~ 216^n / (Pi*n)^2. - _Vaclav Kotesovec_, Jun 21 2021

%t Array[Sum[(3 #)!/(k!*(# - k)!)^3, {k, 0, #}] &, 15, 0] (* _Michael De Vlieger_, Dec 02 2021 *)

%o (PARI) {a(n) = sum(k=0, n, (3*n)!/(k!*(n-k)!)^3)}

%Y Column 3 of A306641.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 02 2019

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Last modified February 22 08:27 EST 2024. Contains 370240 sequences. (Running on oeis4.)