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A325289 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n*(1+x)^(n^2).

%I A325289 #12 Jul 03 2019 20:46:23
%S A325289 1,1,1,3,9,41,200,1096,6440,40095,262298,1790395,12699751,93311273,
%T A325289 708519038,5549751855,44780255681,371785828813,3173019719939,
%U A325289 27813799706468,250222091088035,2308676057468240,21831456961064288,211449264040904335,2096345122112307560,21261235260097878478,220457711039776064974,2335722548273384751833
%N A325289 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n*(1+x)^(n^2).
%H A325289 Paul D. Hanna, <a href="/A325289/b325289.txt">Table of n, a(n) for n = 0..250</a>
%e A325289 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 9*x^4 + 41*x^5 + 200*x^6 + 1096*x^7 + 6440*x^8 + 40095*x^9 + 262298*x^10 + 1790395*x^11 + 12699751*x^12 + ...
%e A325289 such that the following series are equal:
%e A325289 B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + ...
%e A325289 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^8*(1+x)^64 + ...
%e A325289 where
%e A325289 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 A325289 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A325289 A[#A] = -polcoeff( sum(m=0,#A, x^m*( Ser(A)^(m*(m+1)/2) - (1+x +x*O(x^#A))^(m^2)) ),#A) );A[n+1]}
%o A325289 for(n=0,30,print1(a(n),", "))
%Y A325289 Cf. A121689, A325294, A326424.
%K A325289 nonn
%O A325289 0,4
%A A325289 _Paul D. Hanna_, Apr 25 2019