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A200717 G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^2).

%I A200717 #23 Sep 20 2013 06:00:20
%S A200717 1,1,4,18,93,521,3073,18806,118297,760162,4968480,32928392,220766739,
%T A200717 1494635330,10203884795,70167751762,485574854049,3379064343829,
%U A200717 23631314301088,165998001901786,1170706810318259,8286253163771045,58842370488310336,419102145275264242,2993221125640617827
%N A200717 G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^2).
%C A200717 More generally, for fixed parameters p and q, if F(x) satisfies:
%C A200717 F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ), then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).
%H A200717 Vincenzo Librandi, <a href="/A200717/b200717.txt">Table of n, a(n) for n = 0..200</a>
%H A200717 Vaclav Kotesovec, <a href="/A200717/a200717.txt">Recurrence (of order 9)</a>
%F A200717 G.f. A(x) satisfies:
%F A200717 (1) A(x) = (1/x)*Series_Reversion( x*(1 + sqrt(1 - 4*x*(1+x^2)^2)) / (2*(1+x^2)) ).
%F A200717 (2) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [Sum_{k=0..n} C(n,k)^2 * x^k / A(x)^k] ).
%F A200717 (3) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [(1-x/A(x)^2)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k / A(x)^k] ).
%F A200717 a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 7.60435909657327146... is the root of the equation -108 + 27*d^2 + 1620*d^3 - 216*d^4 - 1456*d^5 - 2556*d^6 - 716*d^7 + 20*d^8 + 16*d^9 = 0 and c = 0.45780648099092640511434469483084555191269495951... - _Vaclav Kotesovec_, Sep 19 2013
%e A200717 G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 93*x^4 + 521*x^5 + 3073*x^6 +...
%e A200717 Related expansions:
%e A200717 A(x)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 238*x^4 + 1372*x^5 + 8256*x^6 +...
%e A200717 A(x)^3 = 1 + 3*x + 15*x^2 + 79*x^3 + 447*x^4 + 2655*x^5 + 16324*x^6 +...
%e A200717 A(x)^5 = 1 + 5*x + 30*x^2 + 180*x^3 + 1110*x^4 + 7006*x^5 + 45075*x^6 +...
%e A200717 where A(x) = 1 + x*A(x)^3 + x^2*A(x)^2 + x^3*A(x)^5.
%e A200717 The logarithm of the g.f. A = A(x) equals the series:
%e A200717 log(A(x)) = (1 + x/A)*x*A^2 + (1 + 2^2*x/A + x^2/A^2)*x^2*A^4/2 +
%e A200717 (1 + 3^2*x/A + 3^2*x^2/A^2 + x^3/A^3)*x^3*A^6/3 +
%e A200717 (1 + 4^2*x/A + 6^2*x^2/A^2 + 4^2*x^3/A^3 + x^4/A^4)*x^4*A^8/4 +
%e A200717 (1 + 5^2*x/A + 10^2*x^2/A^2 + 10^2*x^3/A^3 + 5^2*x^4/A^4 + x^5/A^5)*x^5*A^10/5 + ...
%t A200717 nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF^3) * (1 + x^2*AGF^2) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 19 2013 *)
%o A200717 (PARI) {a(n)=polcoeff((1/x)*serreverse( x*(1 + sqrt(1 - 4*x*(1+x^2)^2 +x*O(x^n))) / (2*(1+x^2)) ),n)}
%o A200717 for(n=0,30,print1(a(n),", "))
%o A200717 (PARI) {a(n)=local(p=2,q=-1,A=1+x);for(i=1,n,A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n));polcoeff(A,n)}
%o A200717 (PARI) {a(n)=local(p=2,q=-1,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0,m,binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
%o A200717 (PARI) {a(n)=local(p=2,q=-1,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
%Y A200717 Cf. A200716, A200718, A200719, A200074, A200075, A215576, A199876, A199874, A199877, A198951, A198953, A198957, A192415, A198888, A036765.
%K A200717 nonn
%O A200717 0,3
%A A200717 _Paul D. Hanna_, Nov 21 2011