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%I #13 Sep 28 2023 05:29:21
%S 1,1,3,9,32,118,460,1844,7587,31804,135433,584092,2546250,11201310,
%T 49663816,221701658,995621590,4494862920,20388491423,92872814115,
%U 424665159560,1948516758192,8968647197842,41399782218408,191608577837136
%N G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^n/(1 - x^n*A(x)^n).
%H Vaclav Kotesovec, <a href="/A192206/b192206.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies: A(x) = 1 + Sum_{n>=1} tau(n)*x^n*A(x)^n, where tau(n) = the number of divisors of n (A000005).
%F G.f. satisfies: G(x) = A(x/G(x)) where G(x) = 1 + Sum_{n>=1} x^n/(1 - x^n) is a g.f. for A000005.
%F a(n) ~ c * d^n / n^(3/2), where d = 4.92231207163377242376... and c = 0.58002108600295682... - _Vaclav Kotesovec_, Sep 28 2023
%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 118*x^5 + 460*x^6 +...
%e which satisfies:
%e A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(1-x^2*A(x)^2) + x^3*A(x)^3/(1-x^3*A(x)^3) +...
%t nmax = 30; A[_] = 0; Do[A[x_] = 1 + Sum[x^k*A[x]^k/(1 - x^k*A[x]^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Sep 28 2023 *)
%t (* Calculation of constant d: *) val = r /. FindRoot[{Log[1 - r*s] + QPolyGamma[0, 1, r*s] == (s-1)* Log[r*s], r*(1/(-1 + r*s) + Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]) == Log[r*s] + (s-1)/s}, {r, 1/5}, {s, 2}, WorkingPrecision -> 30] // Quiet; N[1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* _Vaclav Kotesovec_, Sep 28 2023 *)
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^m/(1-x^m*A^m+x*O(x^n)))); polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, sigma(m,0)*x^m*A^m+x*O(x^n))); polcoeff(A, n)}
%Y Cf. A000005, A307397, A307399.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 25 2011
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