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%I #9 Nov 05 2014 16:43:11
%S 1,5,37,325,3125,31925,339077,3700645,41200981,465736725,5328229797,
%T 61552244485,716791570549,8403794763125,99096946864325,
%U 1174370518273125,13977636401394069,167001257979441365,2002052157653251557,24073717683854557125,290261630170911545525,3508332484300450371125
%N G.f.: Sum_{n>=0} 4^n * x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2.
%C a(n) == 5 (mod 16) for n>=1.
%F G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k]^2.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 4^k * Sum_{j=0..k} C(k,j)^2 * 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 * 4^j * x^j.
%F a(n) = Sum_{k=0..[n/2]} 4^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 4^j.
%F Recurrence: (n-3)*n^2*a(n) = 5*(n-3)*(3*n^2 - 3*n + 1)*a(n-1) - (n-1)*(23*n^2 - 92*n + 65)*a(n-2) - 5*(n-2)*(15*n^2 - 60*n + 53)*a(n-3) - 4*(n-3)*(23*n^2 - 92*n + 65)*a(n-4) + 80*(n-1)*(3*n^2 - 21*n + 37)*a(n-5) - 64*(n-4)^2*(n-1)*a(n-6). - _Vaclav Kotesovec_, Nov 05 2014
%F a(n) ~ ((13+3*sqrt(17))/2)^(n+1) / (8*Pi*n). - _Vaclav Kotesovec_, Nov 05 2014
%e G.f.: A(x) = 1 + 5*x + 37*x^2 + 325*x^3 + 3125*x^4 + 31925*x^5 +...
%e where
%e A(x) = 1/(1-x) + 4*x/(1-x)^3*(1+x)^2
%e + 4^2*x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
%e + 4^3*x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2
%e + 4^4*x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
%e + 4^5*x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
%e + 4^6*x^6/(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 +...
%t Table[Sum[4^k * 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_, Nov 05 2014 *)
%o (PARI) /* By definition: */
%o {a(n)=local(A=1); A=sum(m=0, n, 4^m*x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a binomial identity: */
%o {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-4*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 4^k * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a binomial identity: */
%o {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 4^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))), n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a binomial identity: */
%o {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 4^(m-k) * sum(j=0, k, binomial(k, j)^2 * 4^j * x^j)+x*O(x^n))), n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a formula for a(n): */
%o {a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 4^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 4^j))}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A246538, A246539, A246056, A246423, A243948, A245929, A227845, A245925.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 28 2014
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