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%I #20 Sep 30 2023 12:17:06
%S 1,2,4,11,33,111,395,1461,5565,21654,85694,343819,1395221,5716363,
%T 23613391,98238069,411241397,1730976298,7321464729,31102459844,
%U 132645440213,567712900716,2437610016301,10497289161264,45327443941706,196210780501951,851293469453768
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)).
%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*A(x), and r = x.
%H Vaclav Kotesovec, <a href="/A340329/b340329.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) satisfies the following formulas.
%F (1) A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)).
%F (2) A(x) = Sum_{n>=0} x^n*A(x)^n / (1 - x^(n+1)).
%F (3) A(x) = Sum_{n>=0} x(n^2+2*n) * A(x)^n * (1 - x^(2*n+2)*A(x)) / ((1 - x^(n+1))*(1 - x^(n+1)*A(x))). - _Paul D. Hanna_, Feb 20 2021
%F a(n) ~ c * d^n / n^(3/2), where d = 4.5952034330689668887505654340441530851246660208... and c = 0.7049561865662193122992597796630536353257... - _Vaclav Kotesovec_, Jan 07 2021
%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 11*x^3 + 33*x^4 + 111*x^5 + 395*x^6 + 1461*x^7 + 5565*x^8 + 21654*x^9 + 85694*x^10 + ...
%e where
%e A(x) = 1/(1 - x*A(x)) + x/(1 - x^2*A(x)) + x^2/(1 - x^3*A(x)) + x^3/(1 - x^4*A(x)) + x^4/(1 - x^5*A(x)) + ...
%e also
%e A(x) = 1/(1 - x) + x*A(x)/(1 - x^2) + x^2*A(x)^2/(1 - x^3) + x^3*A(x)^3/(1 - x^4) + x^4*A(x)^4/(1 - x^5) + ...
%t (* Calculation of constants {d,c}: *) eq = FindRoot[{(Log[1-r] + QPolyGamma[0, 2 + Log[s]/Log[r], r])/(r* s*Log[r]) == s - 1/(1 - r*s), r/(-1 + r*s)^2 + QPolyGamma[1, 2 + Log[s]/Log[r], r]/(r*s^2*Log[r]^2) == 2 - 1/(s*(1 - r*s))}, {r, 1/4}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r/.eq, 120], val = Sqrt[-r * s^2 * (-1 + r*s) * Log[r]^2*((1 + s*(-1 + s + r*(-1 - (-1 + r)*s^2*(-2 + r*s))) + (-1 + r) * s*(s*(-1 + r*s)^2*Log[r] - (-1 + 2*s*(-1 + r*s)^2) * Log[r*s]) + (-1 + r)*(-1 + r*s)^2* Derivative[0, 0, 1][QPolyGamma][0, 2 + Log[s]/Log[r], r])/(2*Pi*(-1 + r) * (r*s*(1 + s*(-4 + r + 4*r*s*(3 + r*s*(-3 + r*s)))) * Log[r]^3 - (-1 + r*s)^3*QPolyGamma[2, 2 + Log[s]/Log[r], r])))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* _Vaclav Kotesovec_, Sep 30 2023 *)
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m / (1 - x^(m+1)*A +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*A^m / (1 - x^(m+1) +x*O(x^n)) )); ; polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340338, A340355,
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 05 2021
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