%I #15 Jan 09 2018 14:29:48

%S 1,1,5,31,222,1727,14179,120930,1060992,9514463,86818391,803516167,

%T 7524700644,71169939341,678877680077,6523424076116,63087757216084,

%U 613575943566436,5997490784042496,58886692596764215,580516324380845804,5743718741275361697,57017511243375535969

%N G.f. satisfies: A(x) = (1 + x*A(x)^4)*(1 + x^2*A(x)^4).

%H Vaclav Kotesovec, <a href="/A199877/b199877.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k] * x^n*A(x)^(3*n)/n ).

%F (2) A(x) = exp( Sum_{n>=1} [(1-x)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k )] * x^n*A(x)^(3*n)/n.

%F a(n) ~ s * sqrt((1 + 2*r + 3*r^2*s^4) / (2*Pi*(3 + 3*r + 14*r^2*s^4))) / (2*n^(3/2)*r^n), where r = 0.0940387024218615638441791629908854357421782432118... and s = 1.322930427586092521664829345633697493713415726621... are real roots of the system of equations 1 + r*(1 + r)*s^4 + r^3*s^8 = s, 4*r*s^3*(1 + r + 2*r^2*s^4) = 1. - _Vaclav Kotesovec_, Nov 22 2017

%e G.f.: A(x) = 1 + x + 5*x^2 + 31*x^3 + 222*x^4 + 1727*x^5 + 14179*x^6 +...

%e Related expansions:

%e A(x)^4 = 1 + 4*x + 26*x^2 + 188*x^3 + 1471*x^4 + 12124*x^5 + 103684*x^6 +...

%e A(x)^8 = 1 + 8*x + 68*x^2 + 584*x^3 + 5122*x^4 + 45792*x^5 + 416196*x^6 +...

%e where A(x) = 1 + x*(1+x)*A(x)^4 + x^3*A(x)^8.

%e The logarithm of the g.f. equals the series:

%e log(A(x)) = (1 + x)*x*A(x)^3 + (1 + 2^2*x + x^2)*x^2*A(x)^6/2 +

%e (1 + 3^2*x + 3^2*x^2 + x^3)*x^3*A(x)^9/3 +

%e (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^4*A(x)^12/4 +

%e (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)*x^5*A(x)^15/5 +

%e (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^6*A(x)^18/6 +...

%e more explicitly,

%e log(A(x)) = x + 9*x^2/2 + 79*x^3/3 + 733*x^4/4 + 7006*x^5/5 + 68229*x^6/6 + 673268*x^7/7 +...

%t terms = 23; A[_] = 1; Do[A[x_] = (1 + x*A[x]^4)*(1 + x^2*A[x]^4) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 09 2018 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(1 + x*A^4)*(1 + x^2*A^4)+x*O(x^n));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j)*(x*A^3+x*O(x^n))^m/m))); polcoeff(A, n, x)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j)*x^m*A^(3*m)/m))); polcoeff(A, n, x)}

%Y Cf. A199874, A199876.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 11 2011