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A337850 G.f. A(x) satisfies: Sum_{n>=0} x^(n^2) * A(x)^n = Sum_{n>=0} x^n * (1+x)^(n^2).

%I A337850 #11 Oct 06 2020 19:38:50
%S A337850 1,2,5,15,53,217,973,4735,24686,137026,805273,4986541,32409056,
%T A337850 220327688,1562196894,11522725486,88215618320,699573288808,
%U A337850 5736354452771,48556636776122,423674461139747,3805446588538974,35142868684678717,333303148345306269,3243121812554272131
%N A337850 G.f. A(x) satisfies: Sum_{n>=0} x^(n^2) * A(x)^n = Sum_{n>=0} x^n * (1+x)^(n^2).
%H A337850 Paul D. Hanna, <a href="/A337850/b337850.txt">Table of n, a(n) for n = 0..200</a>
%e A337850 G.f.: A(x) = 1 + 2*x + 5*x^2 + 15*x^3 + 53*x^4 + 217*x^5 + 973*x^6 + 4735*x^7 + 24686*x^8 + 137026*x^9 + 805273*x^10 + ...
%e A337850 such that the following series are equal
%e A337850 B(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 + x^25*A(x)^5 + x^36*A(x)^6 + x^49*A(x)^7 + ... + x^(n^2)*A(x)^n + ...
%e A337850 B(x) = 1 + x*(1+x) + x^2*(1+x)^4 + x^3*(1+x)^9 + x^4*(1+x)^16 + x^5*(1+x)^25 + x^6*(1+x)^36 + x^7*(1+x)^49 + ... + x^n*(1+x)^(n^2) + ...
%e A337850 where
%e A337850 B(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 57*x^5 + 231*x^6 + 1023*x^7 + 4926*x^8 + 25483*x^9 + 140601*x^10 + ... + A121689(n)*x^n + ...
%o A337850 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -polcoeff( sum(m=0, #A, x^(m^2)*Ser(A)^m - x^m*(1+x +x*O(x^#A) )^(m^2) ), #A) ); A[n+1]}
%o A337850 for(n=0, 35, print1(a(n), ", "))
%Y A337850 Cf. A325298, A121689.
%K A337850 nonn
%O A337850 0,2
%A A337850 _Paul D. Hanna_, Sep 26 2020