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A246570 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.

%I A246570 #9 Sep 02 2014 03:31:14
%S A246570 1,2,15,116,1001,9322,89363,881376,8860677,90407666,933482527,
%T A246570 9731366060,102259648701,1081810639970,11510355762339,123077391281248,
%U A246570 1321739147949829,14248409211657754,154118033900091139,1672053762899099700,18189628173538580233,198362957005290443978
%N A246570 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 A246570 A bisection of A246563.
%C A246570 Self-convolution of A246572.
%H A246570 Vaclav Kotesovec, <a href="/A246570/a246570.txt">Recurrence (of order 8)</a>
%F A246570 a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-k-j,k)^2 * C(k,j)^2.
%e A246570 G.f.: A(x) = 1 + 2*x + 15*x^2 + 116*x^3 + 1001*x^4 + 9322*x^5 + 89363*x^6 +...
%e A246570 where
%e A246570 A(x) = 1/(1-x) + x/(1-x)^5 * (1 + 2^2*x + x^2)^2
%e A246570 + x^2/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
%e A246570 + 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 A246570 The square-root of the g.f. is an integer series:
%e A246570 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 A246570 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 A246570 (PARI) /* By definition: */
%o A246570 {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 A246570 for(n=0, 25, print1(a(n), ", "))
%o A246570 (PARI) /* From a formula for a(n): */
%o A246570 {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 A246570 for(n=0, 25, print1(a(n), ", "))
%Y A246570 Cf. A246563, A246571, A246572, A246573.
%K A246570 nonn
%O A246570 0,2
%A A246570 _Paul D. Hanna_, Aug 30 2014