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a(n) = binomial(n^2 + 3*n, n).
2

%I #13 Mar 04 2022 01:58:28

%S 1,4,45,816,20475,658008,25827165,1198774720,64276915527,

%T 3911395881900,266401260897200,20082459351180240,1660305826125766950,

%U 149389005978091284720,14533945899753270066525,1520398315196482557890304,170190601112537814791748255

%N a(n) = binomial(n^2 + 3*n, n).

%C Equals the central terms of triangle A214398.

%H Robert Israel, <a href="/A214400/b214400.txt">Table of n, a(n) for n = 0..336</a>

%F a(n) = [x^n] 1/(1 - x)^((n+1)^2). - _Ilya Gutkovskiy_, Oct 04 2017

%F a(n) ~ n^(n-1/2)*exp(n+5/2)/sqrt(2*Pi). - _Robert Israel_, Mar 04 2022

%p seq(binomial(n^2+3*n,n),n=0..30); # _Robert Israel_, Mar 04 2022

%o (PARI) a(n)=binomial(n^2+3*n, n)

%Y Cf. A054688, A178325, A214398.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 15 2012