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%I #12 Feb 22 2018 18:16:56
%S 1,1,1,2,5,15,50,181,698,2837,12062,53374,244923,1162536,5697119,
%T 28786266,149814059,802436166,4420515689,25031466730,145616087486,
%U 869760092469,5330945435272,33508699787635,215863606818041
%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2).
%H Paul D. Hanna, <a href="/A107590/b107590.txt">Table of n, a(n) for n = 0..100</a>
%F G.f. A(x) = x/series-reversion(x*F(x)) and thus A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A107591.
%F G.f. A(x)^2 = x/series-reversion(x*G(x)^2) and thus A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A107592.
%F Contribution from _Paul D. Hanna_, Apr 25 2010: (Start)
%F Let A = g.f. A(x), then A satisfies the continued fraction:
%F A = 1/(1- x/(1- (A-1)*x/(1- A^2*x/(1- A*(A^2-1)*x/(1- A^4*x/(1- A^2*(A^3-1)*x/(1- A^6*x/(1- A^3*(A^4-1)*x/(1- ...)))))))))
%F due to an identity of a partial elliptic theta function.
%F (End)
%e A = 1 + x + x^2*A^1 + x^3*A^3 + x^4*A^6 + x^5*A^10 +...
%e = 1 + x + (x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 15*x^7 +...)
%e + (x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 33*x^7 +...)
%e + (x^4 + 6*x^5 + 21*x^6 + 62*x^7 +...)
%e + (x^5 + 10*x^6 + 55*x^7 +...) +...
%e = 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 50*x^6 + 181*x^7 +...
%o (PARI) {a(n) = my(A=1+x+x*O(x^n)); for(k=1,n, A = 1 + sum(j=1,n, x^j * A^(j*(j-1)/2) + x*O(x^n)) ); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A107591, A107592. A155804, A219358.
%K eigen,nonn
%O 0,4
%A _Paul D. Hanna_, May 17 2005
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