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A107594 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2-n).

%I A107594 #16 Nov 06 2019 03:28:40
%S A107594 1,1,1,3,10,42,194,979,5274,30037,179527,1120612,7280750,49120810,
%T A107594 343547469,2487670468,18631824735,144215785791,1152745117570,
%U A107594 9508011730755,80861962283808,708502494881786,6390084112199801,59272034375915217,564899767969587670
%N A107594 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2-n).
%F A107594 G.f. A(x) = x/series-reversion(x*G107595(x)) and thus A(x) = G107595(x/A(x)) where G107595(x) is the g.f. of A107595.
%F A107594 G.f. A(x)^2 = x/series-reversion(x*G107596(x)^2) and thus A(x) = G107596(x/A(x)^2) where G107596(x) is the g.f. of A107596.
%F A107594 From _Paul D. Hanna_, Apr 25 2010: (Start)
%F A107594 Let A = g.f. A(x), then A satisfies the continued fraction:
%F A107594 A = 1/(1- x/(1- (A^2-1)*x/(1- A^4*x/(1- (A^6-A^2)*x/(1- A^8*x/(1- (A^10-A^4)*x/(1- A^12*x/(1- (A^14-A^6)*x/(1- ...)))))))))
%F A107594 due to an identity of a partial elliptic theta function.
%F A107594 (End)
%e A107594 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 42*x^5 + 194*x^6 + 979*x^7 +...
%e A107594 Let A = A(x) then
%e A107594 A = 1 + x*A^0 + x^2*A^2 + x^3*A^6 + x^4*A^12 + x^5*A^20 + x^6*A^30 +...
%e A107594 = 1 + x + (x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 27*x^6 + 110*x^7 +...)
%e A107594 + (x^3 + 6*x^4 + 21*x^5 + 68*x^6 + 240*x^7 + 948*x^8 + 4140*x^9 +...)
%e A107594 + (x^4 + 12*x^5 + 78*x^6 + 388*x^7 + 1737*x^8 + 7632*x^9 +...)
%e A107594 + (x^5 + 20*x^6 + 210*x^7 + 1580*x^8 + 9795*x^9 +...)
%e A107594 + (x^6 + 30*x^7 + 465*x^8 + 5020*x^9 +...) +...
%t A107594 m = 25; A[_] = 0;
%t A107594 Do[A[x_] = 1 + x + Sum[x^k A[x]^(k^2 - k) + O[x]^j, {k, 2, j}], {j, m}];
%t A107594 CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 05 2019 *)
%o A107594 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^(j^2-j)+x*O(x^n)));polcoeff(A,n)}
%o A107594 for(n=0,30,print1(a(n),", "))
%Y A107594 Cf. A107590, A107595, A107596.
%K A107594 eigen,nonn
%O A107594 0,4
%A A107594 _Paul D. Hanna_, May 17 2005