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A134774 G.f.: A(x) = Product_{n>=1} G(x^n,n)^n where G(x,n) = 1 + x*G(x,n)^n.

%I A134774 #13 Sep 25 2024 09:26:09
%S A134774 1,1,3,6,15,26,66,110,253,460,966,1680,3732,6304,13073,23539,47548,
%T A134774 82362,171463,293578,597934,1056830,2105424,3654919,7533609,12915780,
%U A134774 26112978,46033557,92504870,160298673,330468463,568239653,1161488784
%N A134774 G.f.: A(x) = Product_{n>=1} G(x^n,n)^n where G(x,n) = 1 + x*G(x,n)^n.
%F A134774 G.f.: A(x) = exp( Sum_{n>=1} A105862(n)/n*x^n ), where A105862(n) = Sum_{d|n} binomial(n,d)*n/gcd(n,d).
%F A134774 G.f.: A(x) = Product_{n>=1} [ Series_Reversion( x/(1 + x^n) )/x ]^n.
%e A134774 G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 26*x^5 + 66*x^6 +...
%e A134774 G.f.: A(x) = 1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^4 *...
%e A134774 where the functions G(x,n) are g.f.s of well-known sequences:
%e A134774 G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
%e A134774 G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
%e A134774 G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4 ; etc.
%e A134774 Explicitly, the product yielding the g.f. A(x) begins:
%e A134774 A(x) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...
%o A134774 (PARI) a(n)=if(n==0,1, polcoeff(exp(sum(m=1,n, x^m*sumdiv(m,d, binomial(m,d)/gcd(m,d)))),n))
%o A134774 (PARI) a(n)=polcoeff(prod(m=1,n,(1/x*serreverse(x/(1+x^m +x*O(x^n))))^m),n)
%Y A134774 Cf. A105862 (log(A(x))); A056045 (variant); A000108 (Catalan), A001764, A002293.
%K A134774 nonn
%O A134774 0,3
%A A134774 _Paul D. Hanna_, Nov 11 2007