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%I #15 Oct 01 2023 06:12:10
%S 1,2,4,11,35,121,444,1689,6592,26258,106313,436203,1809727,7579202,
%T 31999297,136050472,582002281,2503242025,10818689627,46959246659,
%U 204623676444,894785832949,3925297799901,17270153317728,76187650017660,336934181461844
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^(n+1)).
%H Vaclav Kotesovec, <a href="/A340338/b340338.txt">Table of n, a(n) for n = 0..280</a>
%F G.f. A(x) satisfies the following relations.
%F (1) A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^(n+1)).
%F (2) A(x) = Sum_{n>=0} x^n*A(x)^n / (1 - x^(n+1)*A(x)^n).
%F (3) A(x) = Sum_{n>=0} x^(n^2+2*n) * A(x)^(n^2+n) * (1 - x^(2*n+2)*A(x)^(2*n+1))/((1 - x^(n+1)*A(x)^n)*(1 - x^(n+1)*A(x)^(n+1))).
%F a(n) ~ c * d^n / n^(3/2), where d = 4.6940696906229278555829659... and c = 0.71283897646430285525... - _Vaclav Kotesovec_, Jan 07 2021
%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 11*x^3 + 35*x^4 + 121*x^5 + 444*x^6 + 1689*x^7 + 6592*x^8 + 26258*x^9 + 106313*x^10 + ...
%e where
%e A(x) = 1/(1 - x*A(x)) + x/(1 - x^2*A(x)^2) + x^2/(1 - x^3*A(x)^3) + x^3/(1 - x^4*A(x)^4) + x^4/(1 - x^5*A(x)^5) + ...
%e also
%e A(x) = 1/(1 - x) + x*A(x)/(1 - x^2*A(x)) + x^2*A(x)^2/(1 - x^3*A(x)^2) + x^3*A(x)^3/(1 - x^4*A(x)^3) + x^4*A(x)^4/(1 - x^5*A(x)^4) + ...
%t (* Calculation of constants {d,c}: *) {1/r, Sqrt[-(1-r)*r*s^2*(-1 + r*s)^2*Log[r*s]^5 * (((1 + ((-1 + r)*(1 + s*(-2 + r - (-1 + r)*r*s) - (-1 + r)*s*(-1 + r*s) * Log[r*s]))/((-1 + r*s)*Log[r]))/ (-1 + r)^2 + (s* Derivative[0, 0, 1][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s])/Log[r])/ (2* Pi*(r*Log[r*s]^3*(-2 - 2*s*(-2 + r*s)*(1 + (-1 + r)*r*s) + (-1 + r)*s* Log[r*s]*(-5 - 4*r*s*(-2 + r*s) - (-1 + r*s)^2*Log[r*s])) + (-1 + r)*(-1 + r*s)^2*(Log[r]^2* QPolyGamma[2, 1 + Log[r]/Log[r*s], r*s] + r*s*Log[r*s]^2*(Log[r*s]*(2 + Log[r*s])* Derivative[0, 0, 1][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s] - 2*Log[r]* Derivative[0, 0, 1][QPolyGamma][1, 1 + Log[r]/Log[r*s], r*s] + r*s*Log[r*s]^2* Derivative[0, 0, 2][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s])))))]} /. FindRoot[{1/(1 - r) + (Log[1 - r*s] + QPolyGamma[0, 1 + Log[r]/Log[r*s], r*s])/(r*Log[r*s]) == s, (1/(-1 + r*s) + Derivative[0, 0, 1][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s])/ Log[r*s] == 1 + (s - 1/(1 - r))/(s*Log[r*s]) + (Log[r]/(r*s*Log[r*s]^3))* QPolyGamma[1, 1 + Log[r]/Log[r*s], r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Oct 01 2023 *)
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m / (1 - x^(m+1)*A^(m+1) +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)*A^m +x*O(x^n)) )); ; polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 05 2021
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