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%I #8 Oct 04 2014 04:42:37
%S 1,5,34,265,2219,19490,177119,1651405,15707416,151791375,1485814989,
%T 14697965770,146673432721,1474490991635,14915914368896,
%U 151701887367585,1550083118902041,15903333300738320,163749905809635219,1691449817705302875,17521670544878571584,181972459046153912945
%N G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].
%F G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * x^k].
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 4^(k-j) * x^j.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^j.
%F a(n) = Sum_{k=0..[n/2]} Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 4^j.
%F a(n) ~ (11+3*sqrt(13))^(n+1) / (Pi * n * 2^(n+7/2)). - _Vaclav Kotesovec_, Oct 04 2014
%e G.f.: A(x) = 1 + 5*x + 34*x^2 + 265*x^3 + 2219*x^4 + 19490*x^5 +...
%t Table[Sum[Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 4^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 04 2014 *)
%o (PARI) /* By definition: */
%o {a(n,p=4,q=1)=local(A=1); A=sum(m=0, n, x^m/(1-p*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * q^k * x^k) * sum(k=0, m, binomial(m,k)^2 * p^k * x^k) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n,4,1), ", "))
%o (PARI) /* By a binomial identity: */
%o {a(n,p=4,q=1)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * p^(m-k) * q^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n,4,1), ", "))
%o (PARI) /* By a binomial identity: */
%o {a(n,p=4,q=1)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * p^(k-j) * q^j * x^j)+x*O(x^n))), n)}
%o for(n=0, 25, print1(a(n,4,1), ", "))
%o (PARI) /* By a binomial identity: */
%o {a(n,p=4,q=1)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * p^(m-k) * sum(j=0, k, binomial(k, j)^2 * q^j * x^j)+x*O(x^n))), n)}
%o for(n=0, 25, print1(a(n,4,1), ", "))
%o (PARI) /* Formula for a(n): */
%o {a(n,p=4,q=1)=sum(k=0, n\2, sum(j=0, n-2*k, q^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * p^j))}
%o for(n=0, 25, print1(a(n,4,1), ", "))
%Y Cf. A246510, A248053, A246423, A246455, A246056, A246813.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 30 2014
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