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A326423 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} x^n / (1-x)^(n*(n-1)/2).

%I A326423 #11 Jul 03 2019 20:45:21
%S A326423 1,0,1,1,4,11,39,147,598,2577,11669,55156,270938,1378577,7247494,
%T A326423 39290662,219304105,1258592815,7418414658,44863100701,278117328554,
%U A326423 1765909629266,11475651209600,76267987517000,518046275820877,3593989140928928,25450794447346211,183860936257142088,1354254148649619126,10164913983190913353,77710718331267769117
%N A326423 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n  =  Sum_{n>=0} x^n / (1-x)^(n*(n-1)/2).
%H A326423 Paul D. Hanna, <a href="/A326423/b326423.txt">Table of n, a(n) for n = 0..300</a>
%e A326423 G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 11*x^5 + 39*x^6 + 147*x^7 + 598*x^8 + 2577*x^9 + 11669*x^10 + 55156*x^11 + 270938*x^12 + 1378577*x^13 + 7247494*x^14 + ...
%e A326423 such that the following series are equal
%e A326423 B(x) = 1 + A(x)*x + A(x)^3*x^2 + A(x)^6*x^3 + A(x)^10*x^4 + A(x)^15*x^5 + A(x)^21*x^6 + A(x)^28*x^7 + A(x)^36*x^8 + A(x)^45*x^9 + ...
%e A326423 and
%e A326423 B(x) = 1 + x + x^2/(1-x) + x^3/(1-x)^3 + x^4/(1-x)^6 + x^5/(1-x)^10 + x^6/(1-x)^15 + x^7/(1-x)^21 + x^8/(1-x)^28 + x^9/(1-x)^36 + ...
%e A326423 where
%e A326423 B(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 43*x^6 + 143*x^7 + 510*x^8 + 1936*x^9 + 7775*x^10 + 32869*x^11 + 145665*x^12 + ... + A098569(n-1)*x^n + ...
%o A326423 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A]=polcoeff( sum(m=0,#A, x^m/(1-x +x*O(x^#A))^(m*(m-1)/2) - x^m*Ser(A)^(m*(m+1)/2) ),#A)); A[n+1]}
%o A326423 for(n=0,35,print1(a(n),", "))
%Y A326423 Cf. A098569, A326424, A325294.
%K A326423 nonn
%O A326423 0,5
%A A326423 _Paul D. Hanna_, Jul 03 2019