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A326424 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} (1+x)^(n*(n-1)/2) * x^n.

%I A326424 #12 Jul 03 2019 20:46:38
%S A326424 1,0,1,0,3,4,20,62,251,1002,4295,19086,88369,423957,2104214,10783054,
%T A326424 56969183,309900293,1733790827,9965992962,58801256594,355808106682,
%U A326424 2206237014216,14007443494601,90994768741426,604395083728629,4101881493676885,28426771732773415,201044377117957190,1450195412613951590,10663346917944740350,79885242459500736025
%N A326424 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n  =  Sum_{n>=0} (1+x)^(n*(n-1)/2) * x^n.
%H A326424 Paul D. Hanna, <a href="/A326424/b326424.txt">Table of n, a(n) for n = 0..300</a>
%e A326424 G.f.: A(x) = 1 + x^2 + 3*x^4 + 4*x^5 + 20*x^6 + 62*x^7 + 251*x^8 + 1002*x^9 + 4295*x^10 + 19086*x^11 + 88369*x^12 + 423957*x^13 + 2104214*x^14 + ...
%e A326424 such that the following series are equal
%e A326424 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 A326424 and
%e A326424 B(x) = 1 + x + (1+x)*x^2 + (1+x)^3*x^3 + (1+x)^6*x^4 + (1+x)^10*x^5 + (1+x)^15*x^6 + (1+x)^21*x^7 + (1+x)^28*x^8 + (1+x)^36*x^9 + ...
%e A326424 where
%e A326424 B(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 27*x^6 + 81*x^7 + 262*x^8 + 910*x^9 + 3363*x^10 + 13150*x^11 + 54135*x^12 + ... + A121690(n-1)*x^n + ...
%o A326424 (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 A326424 for(n=0,35,print1(a(n),", "))
%Y A326424 Cf. A121690, A326423, A325289.
%K A326424 nonn
%O A326424 0,5
%A A326424 _Paul D. Hanna_, Jul 03 2019