%I #10 Aug 31 2018 18:43:34
%S 1,2,3,7,16,41,107,286,782,2179,6172,17702,51272,149727,440341,
%T 1303130,3877837,11596602,34832573,105041452,317900632,965240836,
%U 2939479066,8976098663,27478467863,84314278171,259262013763,798802232031,2465713674230,7624219181757,23612883510015,73241919575468
%N G.f. A(x) satisfies: 1-x = Sum_{n>=0} (A(x) + (-x)^n)^n * (-x)^n.
%H Paul D. Hanna, <a href="/A247332/b247332.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies: 1-x = Sum_{n>=1} x^(n*(n-1)) / (1 - A(x)*(-x)^n)^n.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.2506054895139..., c = 1.82859376... . - _Vaclav Kotesovec_, Sep 29 2014
%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 16*x^4 + 41*x^5 + 107*x^6 + 286*x^7 +...
%e where
%e 1-x = 1 - (A(x) - x)*x + (A(x) + x^2)^2*x^2 - (A(x) - x^3)^3*x^3 + (A(x) + x^4)^4*x^4 - (A(x) - x^5)^5*x^5 + (A(x) + x^6)^6*x^6 - (A(x) - x^7)^7*x^7 +-...
%e Also, the g.f. satisfies the series identity:
%e 1-x = 1/(1 + A(x)*x) + x^2/(1 - A(x)*x^2)^2 + x^6/(1 + A(x)*x^3)^3 + x^12/(1 - A(x)*x^4)^4 + x^20/(1 + A(x)*x^5)^5 + x^30/(1 - A(x)*x^6)^6 + x^42/(1 + A(x)*x^7)^7 +...
%o (PARI) {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=Vec(sum(k=0, #A, (Ser(A) + (-x)^k)^k*(-x)^k))[#A+1]);A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=Vec(sum(k=1,sqrtint(#A)+1, x^(k^2-k)/(1 - Ser(A)*(-x)^k)^k ))[#A+1]);A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A317997.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 26 2014