cp's OEIS Frontend

A219263 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x^n)/n / Product_{k>=1} (1 - x^(n*k)*A(x^n)^k) ).

%I A219263 #5 Nov 16 2012 18:35:57
%S A219263 1,1,3,10,39,159,693,3101,14292,67116,320448,1549834,7579037,37406737,
%T A219263 186102602,932294987,4698796087,23809155711,121219100012,619800529792,
%U A219263 3181291257740,16385813881342,84666104373097,438742341955132,2279628504172080,11873579440176774,61984238371422197
%N A219263 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x^n)/n / Product_{k>=1} (1 - x^(n*k)*A(x^n)^k) ).
%C A219263 Compare to the dual g.f. G(x) of A219262:
%C A219263 G(x) = exp( Sum_{n>=1} x^n*G(x)^n/n / Product_{k>=1} (1 - x^(n*k)*G(x^k)^n) ).
%e A219263 G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 159*x^5 + 693*x^6 + 3101*x^7 +...
%e A219263 where
%e A219263 log(A(x)) = x*A(x)/1/((1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)*...) +
%e A219263 x^2*A(x^2)/2/((1-x^2*A(x^2))*(1-x^4*A(x^2)^2)*(1-x^6*A(x^2)^3)*...) +
%e A219263 x^3*A(x^3)/3/((1-x^3*A(x^3))*(1-x^6*A(x^3)^2)*(1-x^9*A(x^3)^3)*...) +
%e A219263 x^4*A(x^4)/4/((1-x^4*A(x^4))*(1-x^8*A(x^4)^2)*(1-x^12*A(x^4)^3)*...) +...
%e A219263 Explicitly,
%e A219263 log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 109*x^4/4 + 531*x^5/5 + 2726*x^6/6 + 13952*x^7/7 + 72581*x^8/8 + 379264*x^9/9 + 1994875*x^10/10 +...
%o A219263 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*subst(A, x, x^m +x*O(x^n))/m/prod(k=1, n\m+1, 1-x^(m*k)*subst(A^k, x, x^m +x*O(x^n)))))); polcoeff(A, n)}
%o A219263 for(n=0, 30, print1(a(n), ", "))
%Y A219263 Cf. A219231, A219261, A219262.
%K A219263 nonn
%O A219263 0,3
%A A219263 _Paul D. Hanna_, Nov 16 2012