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%I #15 Nov 05 2014 14:25:39
%S 1,2,8,36,182,964,5296,29832,171238,997244,5874992,34937400,209392796,
%T 1263258760,7664233696,46726270992,286089651718,1758215706476,
%U 10841476837424,67049791851672,415784950498964,2584585251386296,16101542183281312,100511325748165488,628579719997550044
%N 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 * 2^k * x^k].
%H Paul D. Hanna, <a href="/A243948/b243948.txt">Table of n, a(n) for n = 0..400</a>
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 2^j * 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, k)^2 * 2^k.
%F Recurrence: (n-3)*(n-2)*n^2*a(n) = 2*(n-3)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 2*(n-1)*(5*n^3 - 30*n^2 + 58*n - 38)*a(n-2) + 8*(n-2)*a(n-3) + 4*(n-3)*(5*n^3 - 30*n^2 + 58*n - 34)*a(n-4) - 8*(n-1)*(4*n^3 - 36*n^2 + 106*n - 101)*a(n-5) + 8*(n-4)^2*(n-2)*(n-1)*a(n-6). - _Vaclav Kotesovec_, Aug 17 2014
%F a(n) ~ sqrt(c) * d^n / (2^(3/2)*Pi*n), where d = 2 + sqrt(2) + 2*sqrt(1+sqrt(2)) = 6.52176151043316966349... is the root of the equation 4 - 16*d + 12*d^2 - 8*d^3 + d^4 = 0, and c = 4 + 5/sqrt(2) + 2*sqrt(7+5*sqrt(2)) = 15.0378183078640521... is the root of the equation 1 - 32*c + 60*c^2 - 64*c^3 + 4*c^4 = 0. - _Vaclav Kotesovec_, Aug 17 2014
%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 182*x^4 + 964*x^5 + 5296*x^6 +...
%e where the g.f. is given by the binomial series:
%e A(x) = 1/(1-x) + x/(1-x)^3*(1+x) * (1+2*x)
%e + x^2/(1-x)^5*(1 + 2^2*x + x^2) * (1 + 2^2*2*x + 4*x^2)
%e + x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3)
%e + x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4)
%e + x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5) +...
%e We can also express the g.f. by the binomial series identity:
%e A(x) = 1 + x*(1 + (1+2*x)) + x^2*(1 + 2^2*(1+2*x) + (1+2^2*2*x+4*x^2))
%e + x^3*(1 + 3^2*(1+2*x) + 3^2*(1+2^2*2*x+4*x^2) + (1+3^2*2*x+3^2*4*x^2+8*x^3))
%e + x^4*(1 + 4^2*(1+2*x) + 6^2*(1+2^2*2*x+4*x^2) + 4^2*(1+3^2*2*x+3^2*4*x^2+8*x^3) + (1+4^2*2*x+6^2*4*x^2+4^2*8*x^3+16*x^4))
%e + x^5*(1 + 5^2*(1+2*x) + 10^2*(1+2^2*2*x+4*x^2) + 10^2*(1+3^2*2*x+3^2*4*x^2+8*x^3) + 5^2*(1+4^2*2*x+6^2*4*x^2+4^2*8*x^3+16*x^4) + (1+5^2*2*x+10^2*4*x^2+10^2*8*x^3+5^2*16*x^4+32*x^5)) +...
%o (PARI) /* By definition: */
%o {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*x^k) * sum(k=0, m, binomial(m, k)^2*2^k*x^k) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 30, 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*sum(j=0, k, binomial(k, j)^2*2^j*x^j)+x*O(x^n))), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* Formula for a(n): */
%o {a(n)=sum(k=0, n\2, sum(j=0, n-2*k, binomial(n-k, k+j)^2*binomial(k+j, k)^2*2^k))}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A246455 (dual), A245929, A227845, A245925.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 16 2014
%E Minor edits by _Vaclav Kotesovec_, Nov 05 2014
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