%I #9 Sep 02 2014 03:31:14

%S 1,2,15,116,1001,9322,89363,881376,8860677,90407666,933482527,

%T 9731366060,102259648701,1081810639970,11510355762339,123077391281248,

%U 1321739147949829,14248409211657754,154118033900091139,1672053762899099700,18189628173538580233,198362957005290443978

%N G.f.: Sum_{n>=0} x^n / (1-x)^(4*n+1) * [Sum_{k=0..2*n} C(2*n,k)^2 * x^k]^2.

%C A bisection of A246563.

%C Self-convolution of A246572.

%H Vaclav Kotesovec, <a href="/A246570/a246570.txt">Recurrence (of order 8)</a>

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

%e G.f.: A(x) = 1 + 2*x + 15*x^2 + 116*x^3 + 1001*x^4 + 9322*x^5 + 89363*x^6 +...

%e where

%e A(x) = 1/(1-x) + x/(1-x)^5 * (1 + 2^2*x + x^2)^2

%e + x^2/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2

%e + x^3/(1-x)^13 * (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +...

%e The square-root of the g.f. is an integer series:

%e A(x)^(1/2) = 1 + x + 7*x^2 + 51*x^3 + 425*x^4 + 3879*x^5 + 36527*x^6 + 355333*x^7 + 3531175*x^8 + 35673875*x^9 +...+ A246572(n)*x^n +...

%t Table[Sum[Sum[Binomial[2*n-k-j,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+1) * sum(k=0, 2*m, binomial(2*m, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)}

%o for(n=0, 25, 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), binomial(2*n-k-j, k)^2 * binomial(k, j)^2 ))}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A246563, A246571, A246572, A246573.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 30 2014