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%I #7 Mar 30 2012 18:37:33
%S 1,1,4,19,116,683,4818,31126,232058,1598611,12315375,86887285,
%T 695017086,4999457900,40344295044,298468091712,2434392979661,
%U 18077507384936,150454415661096,1125745880242406,9386869540033292,71518155964958242,597727034006054509
%N G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^3 * 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 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 3*A203267(n*k)*x^(n*k)/k ) ) where A(x) = exp( Sum_{n>=1} A203267(n)*x^n/n ).
%F The logarithmic derivative yields A203267.
%e G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 116*x^4 + 683*x^5 + 4818*x^6 +...
%e G.f.: A(x) = exp( Sum_{n>=1} A203267(n) * x^n/n ),
%e where A(x) = exp( Sum_{n>=1} G_n(x^n)^3 * x^n/n )
%e and G_n(x) = exp( Sum_{k>=1} A203267(n*k)*x^k/k ), which begin:
%e G_1(x) = A(x);
%e G_2(x) = 1 + 7*x + 210*x^2 + 8837*x^3 + 427910*x^4 + 22758491*x^5 +...;
%e G_3(x) = 1 + 46*x + 12280*x^2 + 4087909*x^3 + 1805475734*x^4 +...;
%e G_4(x) = 1 + 371*x + 776202*x^2 + 2360146453*x^3 +...;
%e G_5(x) = 1 + 2611*x + 49859649*x^2 + 1211412677799*x^3 +...;
%e G_6(x) = 1 + 22444*x + 3385662240*x^2 + 742868246890817*x^3 +...;
%e G_7(x) = 1 + 163010*x + 223920974239*x^2 + 396998122840515180*x^3 +...;
%e G_8(x) = 1 + 1414763*x + 15479260324770*x^2 + 249608398400792533605*x^3 +...;
%e ...
%e Also, G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u = n-th root of unity:
%e G_2(x^2) = A(x)*A(-x);
%e G_3(x^3) = A(x)*A(u*x)*A(u^2*x) where u = exp(2*Pi*I/3);
%e 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 ...
%e The logarithmic derivative of this sequence yields A203267:
%e A203267 = [1,7,46,371,2611,22444,163010,1414763,10666423,...].
%o (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), 3*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); polcoeff(exp(x*Ser(vector(n, m, L[m]/m))), n)}
%o (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^3, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)}
%Y Cf. A203267 (log), A203254, A203266.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 30 2011
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