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A192206
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G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^n/(1 - x^n*A(x)^n).
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5
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1, 1, 3, 9, 32, 118, 460, 1844, 7587, 31804, 135433, 584092, 2546250, 11201310, 49663816, 221701658, 995621590, 4494862920, 20388491423, 92872814115, 424665159560, 1948516758192, 8968647197842, 41399782218408, 191608577837136
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OFFSET
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0,3
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LINKS
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FORMULA
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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).
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.
a(n) ~ c * d^n / n^(3/2), where d = 4.92231207163377242376... and c = 0.58002108600295682... - Vaclav Kotesovec, Sep 28 2023
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 118*x^5 + 460*x^6 +...
which satisfies:
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) +...
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MATHEMATICA
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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 *)
(* 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 *)
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PROG
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(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)}
(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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