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A325294 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 A325294 #11 Jul 03 2019 20:45:42
%S A325294 1,1,2,5,17,73,368,2074,12663,82236,561664,4004815,29662508,227413816,
%T A325294 1800063339,14681764890,123207630130,1062547709801,9407762681632,
%U A325294 85445941932906,795514580068247,7587015660017106,74078917658328970,740060483734580171,7560421405484047766,78939580213645975075,841942979579094942598,9168184497787176646141,101876790751549107815492
%N A325294 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 A325294 Paul D. Hanna, <a href="/A325294/b325294.txt">Table of n, a(n) for n = 0..250</a>
%e A325294 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 73*x^5 + 368*x^6 + 2074*x^7 + 12663*x^8 + 82236*x^9 + 561664*x^10 + 4004815*x^11 + 29662508*x^12 + ...
%e A325294 such that the following series are equal:
%e A325294 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 A325294 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 A325294 where
%e A325294 B(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 + 1730*x^7 + 8889*x^8 + 48829*x^9 + 284858*x^10 + 1755325*x^11 + ... + A178325(n)*x^n + ...
%o A325294 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A325294 A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - 1/(1-x +x*O(x^#A))^(m^2)) ), #A) ); A[n+1]}
%o A325294 for(n=0, 30, print1(a(n), ", "))
%Y A325294 Cf. A178325, A325289, A326423.
%K A325294 nonn
%O A325294 0,3
%A A325294 _Paul D. Hanna_, Apr 25 2019