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%I #6 Sep 02 2014 03:23:41
%S 1,6,39,340,3041,28718,279987,2788464,28256709,290124182,3010689527,
%T 31516942060,332347297141,3526399820374,37616896717155,
%U 403127408462816,4337723615579781,46842172878701486,507454305359968827,5513119883595629556,60050379276555861857,655611405802102543086
%N G.f.: Sum_{n>=0} x^n / (1-x)^(4*n+3) * [Sum_{k=0..2*n+1} C(2*n+1,k)^2 * x^k]^2.
%C A bisection of A246563.
%C Self-convolution of A246573.
%H Vaclav Kotesovec, <a href="/A246571/a246571.txt">Recurrence (of order 8)</a>
%F a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-k-j+1,k)^2 * C(k,j)^2.
%e G.f.: A(x) = 1 + 6*x + 39*x^2 + 340*x^3 + 3041*x^4 + 28718*x^5 + 279987*x^6 +...
%e where
%e A(x) = 1/(1-x)^3 * (1 + x)^2 + x/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3)^2
%e + x^2/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
%e + x^3/(1-x)^15 * (1 + 7^2*x + 21^2*x^2 + 35^2*x^3 + 35^2*x^4 + 21^2*x^5 + 7^2*x^6 + x^7)^2 +...
%e The square-root of the g.f. is an integer series:
%e A(x)^(1/2) = 1 + 3*x + 15*x^2 + 125*x^3 + 1033*x^4 + 9385*x^5 + 88531*x^6 + 858739*x^7 + 8517503*x^8 + 85867417*x^9 +...+ A246573(n)*x^n +...
%t Table[Sum[Sum[Binomial[2*n-k-j+1,k]^2 * Binomial[k,j]^2,{j,0,k}],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 02 2014 *)
%o (PARI) /* By definition: */
%o {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(4*m+3) * sum(k=0, 2*m+1, binomial(2*m+1, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) /* From a formula for a(n): */
%o {a(n)=sum(k=0, n, sum(j=0, min(k, 2*n-2*k+1), binomial(2*n-k-j+1, k)^2 * binomial(k, j)^2 ))}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A246563, A246570, A246572, A246573.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 30 2014
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