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%I
%S 1,3,4,7,12,20,52,124,297,802,2114,5705,15653,42392,116560,324503,
%T 907520,2556402,7223284,20471723,58319247,166859181,479305506,
%U 1381683897,3993923929,11574493329,33625052782,97908062011,285724318094,835602499442
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)^n).
%C The g.f. A(x) of this sequence is motivated by the following identity:
%C Sum_{n>=0} C(t+n-1,n) * p^n/(1 - q*r^n)^s = Sum_{n>=0} C(s+n-1,n) * q^n/(1 - p*r^n)^t ;
%C here, p = x, q = x, r = x*A(x), s = 1, and t = 2.
%F G.f. A(x) satisfies the following relations.
%F (1) A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)^n).
%F (2) A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^n)^2.
%e G.f.: A(x) = 1 + 3*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 52*x^6 + 124*x^7 + 297*x^8 + 802*x^9 + 2114*x^10 + 5705*x^11 + 15653*x^12 + ...
%e where
%e A(x) = 1/(1 - x) + 2*x/(1 - x^2*A(x)) + 3*x^2/(1 - x^3*A(x)^2) + 4*x^3/(1 - x^4*A(x)^3) + 5*x^4/(1 - x^5*A(x)^4) + ...
%e also
%e A(x) = 1/(1 - x)^2 + x/(1 - x^2*A(x))^2 + x^2/(1 - x^3*A(x)^2)^2 + x^3/(1 - x^4*A(x)^3)^2 + x^4/(1 - x^5*A(x)^4)^2 + ...
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (m+1) * x^m / (1 - x^(m+1)*A^m +x*O(x^n)) )); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m / (1 - x^(m+1)*A^m +x*O(x^n))^2 )); ; polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340329, A340338, A340355, A340356, A340357, A340358, A340360.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 07 2021
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