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A326089 G.f. A(x) satisfies: 1 + 2 * Sum_{n>=1} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} (1 + x*A(x)^n)^n * x^n.

%I A326089 #3 Jun 28 2019 17:25:08
%S A326089 1,2,10,90,1002,12422,164866,2294842,33092066,490458214,7430814938,
%T A326089 114644436410,1796058433818,28510908961974,457838834794898,
%U A326089 7427992562185162,121633693137277970,2008722124583739830,33435242160622759594,560667079009101397162,9468303108907658924874,160991659768738107957670,2755748012982878460654370,47485639148930596873348890
%N A326089 G.f. A(x) satisfies: 1 + 2 * Sum_{n>=1} A(x)^(n*(n+1)/2) * x^n  =  Sum_{n>=0} (1 + x*A(x)^n)^n * x^n.
%F A326089 The g.f. A(x) allows for the following sums to be equal:
%F A326089 (1) B(x) = 1 + 2 * Sum_{n>=1} A(x)^(n*(n+1)/2) * x^n.
%F A326089 (2) B(x) = Sum_{n>=0} (1 + x*A(x)^n)^n * x^n.
%F A326089 (3) B(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1 - x*A(x)^n)^(n+1).
%e A326089 G.f.: A(x) = 1 + 2*x + 10*x^2 + 90*x^3 + 1002*x^4 + 12422*x^5 + 164866*x^6 + 2294842*x^7 + 33092066*x^8 + 490458214*x^9 + 7430814938*x^10 + ...
%e A326089 such that the following sum
%e A326089 B(x) = 1 + 2*A(x)*x + 2*A(x)^3*x^2 + 2*A(x)^6*x^3 + 2*A(x)^10*x^4 + 2*A(x)^15*x^5 + 2*A(x)^21*x^6 + 2*A(x)^28*x^7 + ... + 2*A(x)^(n*(n+1)/2)*x^n + ...
%e A326089 equals
%e A326089 B(x) = 1 + (1 + x*A(x))*x + (1 + x*A(x)^2)^2*x^2 + (1 + x*A(x)^3)^3*x^3 + (1 + x*A(x)^4)^4*x^4 + (1 + x*A(x)^5)^5*x^5 + ... + (1 + x*A(x)^n)^n*x^n + ...
%e A326089 as well as
%e A326089 B(x) = 1/(1 - x) + A(x)*x/(1 - x*A(x))^2 + A(x)^4*x^2/(1 - x*A(x)^2)^3 + A(x)^9*x^3/(1 - x*A(x)^3)^4 + A(x)^16*x^4/(1 - x*A(x)^4)^5 + ...
%e A326089 where
%e A326089 B(x) = 1 + 2*x + 6*x^2 + 34*x^3 + 290*x^4 + 3082*x^5 + 37078*x^6 + 482122*x^7 + 6611538*x^8 + 94256914*x^9 + 1384318518*x^10 + ...
%o A326089 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = polcoeff( sum(m=0,#A, (1 + Ser(A)^m)^m*x^m - 2*x^m*Ser(A)^(m*(m+1)/2) ),#A)); A[n+1]}
%o A326089 for(n=0,30, print1(a(n),", "))
%K A326089 nonn
%O A326089 0,2
%A A326089 _Paul D. Hanna_, Jun 28 2019