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%I #9 Feb 20 2021 14:42:49
%S 1,1,1,2,5,14,42,133,440,1510,5347,19459,72561,276616,1076236,4268236,
%T 17238623,70858091,296293158,1260044245,5449129205,23962691920,
%U 107160352895,487379459886,2254710459801,10611155135759,50808249311687,247538711398811
%N G.f. A(x) satisfies: A(x) = (1-x) * Sum_{n>=0} x^n / (1 - x*A(x)^n).
%C The g.f. of this sequence is motivated by the following identity:
%C Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
%C here, p = x, q = x, and r = A(x).
%F G.f. A(x) satisfies:
%F (1) A(x) = (1-x) * Sum_{n>=0} x^n / (1 - x*A(x)^n).
%F (2) A(x) = (1-x) * Sum_{n>=0} x^(2*n) * A(x)^(n^2) * (1 + x*A(x)^n) / (1 - x*A(x)^n). - _Paul D. Hanna_, Feb 20 2021
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 133*x^7 + 440*x^8 + 1510*x^9 + 5347*x^10 + 19459*x^11 + 72561*x^12 + ...
%e where
%e A(x)/(1-x) = 1/(1-x) + x/(1 - x*A(x)) + x^2/(1 - x*A(x)^2) + x^3/(1 - x*A(x)^3) + x^4/(1 - x*A(x)^4) + x^5/(1 - x*A(x)^5) + ...
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = (1-x) * sum(m=0,n, x^m / (1 - x*A^m +x*O(x^n)) ) ); polcoeff(H=A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340891.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 25 2021
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