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A206847 a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).

%I A206847 #14 Jan 28 2019 06:48:38
%S A206847 1,2,18,2270,3678482,51789416252,9723940840418814,
%T A206847 13783866167176942874214,260749663122506218247699587346,
%U A206847 35385577627626083328957267246097557212,64138056102285851525440919122006580387539950268,814449089808478655249485968539593253265395820497817710866
%N A206847 a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).
%C A206847 Ignoring the initial term a(0), equals the logarithmic derivative of A206846.
%H A206847 Seiichi Manyama, <a href="/A206847/b206847.txt">Table of n, a(n) for n = 0..45</a>
%F A206847 From _Vaclav Kotesovec_, Mar 03 2014: (Start)
%F A206847 Limit n->infinity a(n)^(1/n^2) = 16/(3*sqrt(3)).
%F A206847 a(n) ~ c * 2^(4*n^2+3) / (Pi * n^2 * 3^(3*n^2/2+1)), where c = JacobiTheta3(0,9*exp(-16/3)) = EllipticTheta[3, 0, 9*Exp[-16/3]] = 1.08691022925895131... if n is even, and c = JacobiTheta2(0,9*exp(-16/3)) = EllipticTheta[2, 0, 9*Exp[-16/3]] = 0.91485129628884995... if n is odd.
%F A206847 (End)
%e A206847 L.g.f.: L(x) = 2*x + 18*x^2/2 + 2270*x^3/3 + 3678482*x^4/4 + 51789416252*x^5/5 +...
%e A206847 where exponentiation yields the g.f. of A206846:
%e A206847 exp(L(x)) = 1 + 2*x + 11*x^2 + 776*x^3 + 921193*x^4 + 10359730908*x^5 +...
%e A206847 Illustration of terms: by definition,
%e A206847 a(1) = C(1,0)*C(1,1) + C(1,1)*C(1,0);
%e A206847 a(2) = C(4,0)*C(4,4) + C(4,1)*C(4,1) + C(4,4)*C(4,0);
%e A206847 a(3) = C(9,0)*C(9,9) + C(9,1)*C(9,4) + C(9,4)*C(9,1) + C(9,9)*C(9,0);
%e A206847 a(4) = C(16,0)*C(16,16) + C(16,1)*C(16,9) + C(16,4)*C(16,4) + C(16,9)*C(16,1) + C(16,16)*C(16,0); ...
%e A206847 Numerically, the above evaluates to be:
%e A206847 a(1) = 1*1 + 1*1 = 2;
%e A206847 a(2) = 1*1 + 4*4 + 1*1 = 18;
%e A206847 a(3) = 1*1 + 9*126 + 126*9 + 1*1 = 2270;
%e A206847 a(4) = 1*1 + 16*11440 + 1820*1820 + 11440*16 + 1*1 = 3678482;
%e A206847 a(5) = 1*1 + 25*2042975 + 12650*2042975 + 2042975*12650 + 2042975*25 + 1*1 = 51789416252; ...
%t A206847 Table[Sum[Binomial[n^2, k^2] * Binomial[n^2, (n-k)^2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)
%o A206847 (PARI) {a(n)=sum(k=0, n, binomial(n^2,(n-k)^2)*binomial(n^2,k^2))}
%o A206847 for(n=1, 20, print1(a(n), ", "))
%Y A206847 Cf. A206846 (exp), A206849, A206851.
%K A206847 nonn
%O A206847 0,2
%A A206847 _Paul D. Hanna_, Feb 15 2012