A229052 - OEIS

%I #13 Jul 22 2014 06:20:38

%S 1,2,6,92,6662,2150552,3093730764,18251332286098,466740831542894470,

%T 47238803741195397513182,20522607409110459026633535856,

%U 34700017072200465774261952422246668,250699892545838622857396499800167790109260,6984916990466628202550631436961441381064765905022

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

%H Vincenzo Librandi, <a href="/A229052/b229052.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = Sum_{k=0..n} binomial(n^2-n*k, (n-k)^2) * binomial(n*k, k^2).

%F a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k).

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

%F a(n) ~ c * 2^(n^2+2)/(Pi*n^2), where c = EllipticTheta[3,0,1/E^2] = 1.271341522189... if n is even and c = EllipticTheta[2,0,1/E^2] = 1.23528676585389... if n is odd. - _Vaclav Kotesovec_, Sep 22 2013

%e The triangle A228832(n,k) = C(n*k, k^2) illustrates the terms involved in the sum a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k):

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 15, 1;

%e 1, 4, 70, 220, 1;

%e 1, 5, 210, 5005, 4845, 1;

%e 1, 6, 495, 48620, 735471, 142506, 1;

%e 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1; ...

%t Table[Sum[Binomial[n^2 - n k, n k - k^2] Binomial[n k, k^2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 22 2013 *)

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

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A228832, A206847, A218792.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 22 2013