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A225602 a(n) = A002426(n^2), where A002426 is the central trinomial coefficients.

%I A225602 #13 Feb 28 2017 22:35:50
%S A225602 1,1,19,3139,5196627,82176836301,12159131877715993,
%T A225602 16639279789182494873661,209099036316263774148543463251,
%U A225602 24017537903429183163390175566336055657,25134265191388162956642519120384003897467908119,239089990313305548946878880624659134220897530949847409821
%N A225602 a(n) = A002426(n^2), where A002426 is the central trinomial coefficients.
%H A225602 G. C. Greubel, <a href="/A225602/b225602.txt">Table of n, a(n) for n = 0..45</a>
%F A225602 Logarithmic derivative of A225604 (ignoring the initial term of this sequence).
%F A225602 a(n) = Sum_{k=0..floor(n^2/2)} binomial(n^2, k) * binomial(n^2-k, k).
%e A225602 L.g.f.: L(x) = x + 19*x^2/2 + 3139*x^3/3 + 5196627*x^4/4 + 82176836301*x^5/5 + ...
%e A225602 where exponentiation is an integer series:
%e A225602 exp(L(x)) = 1 + x + 10*x^2 + 1056*x^3 + 1300253*x^4 + 16436676927*x^5 + ... + A225604(n)*x^n + ...
%t A225602 Table[Sum[Binomial[n^2, k]*Binomial[n^2 - k, k], {k, 0, Floor[n^2/2]}], {n,0,50}] (* _G. C. Greubel_, Feb 27 2017 *)
%o A225602 (PARI) {a(n)=sum(k=0, n^2\2, binomial(n^2, k)*binomial(n^2-k, k))}
%o A225602 for(n=0, 20, print1(a(n), ", "))
%Y A225602 Cf. A225604, A002426.
%K A225602 nonn
%O A225602 0,3
%A A225602 _Paul D. Hanna_, Aug 03 2013