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A203266 G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u is an n-th root of unity.

%I A203266 #7 Mar 30 2012 18:37:33
%S A203266 1,1,3,10,43,172,852,3719,19290,90469,481825,2295973,12812880,
%T A203266 62122518,346770241,1744884177,9830723932,49268101457,285020577850,
%U A203266 1438021808125,8300553742452,42955378420263,247456097485363,1271978597532857,7520668963944205,38700952324615078
%N A203266 G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u is an n-th root of unity.
%F A203266 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*A203265(n*k)*x^(n*k)/k ) ) where A(x) = exp( Sum_{n>=1} A203265(n)*x^n/n ).
%F A203266 The logarithmic derivative yields A203265.
%e A203266 G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 + 3719*x^7 +...
%e A203266 G.f.: A(x) = exp( Sum_{n>=1} A203265(n) * x^n/n ),
%e A203266 where A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n )
%e A203266 and G_n(x) = exp( Sum_{k>=1} A203265(n*k)*x^k/k ), which begin:
%e A203266 G_1(x) = A(x);
%e A203266 G_2(x) = 1 + 5*x + 75*x^2 + 1518*x^3 + 34663*x^4 + 867760*x^5 +...;
%e A203266 G_3(x) = 1 + 22*x + 2019*x^2 + 214648*x^3 + 31221037*x^4 +...;
%e A203266 G_4(x) = 1 + 125*x + 59771*x^2 + 40659310*x^3 + 31438395303*x^4 +...;
%e A203266 G_5(x) = 1 + 576*x + 1760688*x^2 + 6380121685*x^3 +...;
%e A203266 G_6(x) = 1 + 3554*x + 57073923*x^2 + 1295238092004*x^3 +...;
%e A203266 G_7(x) = 1 + 16843*x + 1719312892*x^2 + 212162358939394*x^3 +...;
%e A203266 G_8(x) = 1 + 103917*x + 56284535547*x^2 + 44125115136389518*x^3 +...;
%e A203266 ...
%e A203266 Also, G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u = n-th root of unity:
%e A203266 G_2(x^2) = A(x)*A(-x);
%e A203266 G_3(x^3) = A(x)*A(u*x)*A(u^2*x) where u = exp(2*Pi*I/3);
%e A203266 G_4(x^4) = A(x)*A(u*x)*A(u^2*x)*A(u^3*x) where u = exp(2*Pi*I/4);
%e A203266 ...
%e A203266 The logarithmic derivative of this sequence yields A203265:
%e A203266 A203265 = [1,5,22,125,576,3554,16843,103917,521338,3189600,...].
%o A203266 (PARI) {a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), 2*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); polcoeff(exp(x*Ser(vector(n, m, L[m]/m))), n)}
%o A203266 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, x^m/m*round(prod(k=0, m-1, subst(A^2, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)}
%Y A203266 Cf. A203265 (log), A203254, A203268.
%K A203266 nonn
%O A203266 0,3
%A A203266 _Paul D. Hanna_, Dec 30 2011