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A336522 a(n) is the coefficient of x^(n^2) in expansion of ( (1 + x)/(1 - x) )^n.

%I A336522 #38 Aug 09 2025 20:55:07
%S A336522 1,2,16,326,11008,525002,32497680,2478629134,224921989120,
%T A336522 23681262354194,2838826197080080,381825269929428822,
%U A336522 56949892477659339520,9329658433405643973850,1665421971238565711337488,321771059958076157377283102,66901218825369170336327860224,14894388013750938445628478094370
%N A336522 a(n) is the coefficient of x^(n^2) in expansion of ( (1 + x)/(1 - x) )^n.
%H A336522 Seiichi Manyama, <a href="/A336522/b336522.txt">Table of n, a(n) for n = 0..311</a>
%F A336522 a(n) = (1/n) * [x^n] ( (1 + x)/(1 - x) )^(n^2) for n > 0.
%F A336522 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k-1,n-1).
%F A336522 a(n) = (1/n) * Sum_{k=0..n} binomial(n^2,n-k) * binomial(n^2+k-1,k) for n > 0.
%F A336522 a(n) = Sum_{k=1..n} 2^k * binomial(n,k) * binomial(n^2-1,k-1) for n > 0.
%F A336522 a(n) ~ 2^(n - 1/2) * exp(n) * n^(n - 3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Jul 31 2021
%F A336522 a(n) = binomial(n^2-1, n-1)*hypergeom([-n, n^2], [1-n+n^2], -1). - _Stefano Spezia_, Aug 09 2025
%t A336522 a[n_] := Sum[Binomial[n, k] * Binomial[n^2 + k - 1, n - 1], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Jul 24 2020 *)
%o A336522 (PARI) {a(n) = if(n==0, 1, sum(k=0, n, binomial(n^2, n-k) * binomial(n^2+k-1, k))/n)}
%o A336522 (PARI) {a(n) = if(n==0, 1, sum(k=1, n, 2^k*binomial(n, k) * binomial(n^2-1, k-1)))}
%Y A336522 Main diagonal of A336521.
%Y A336522 Cf. A336537.
%K A336522 nonn
%O A336522 0,2
%A A336522 _Seiichi Manyama_, Jul 24 2020