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A190861
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G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x))/(1-x^n).
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2
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1, 2, 6, 18, 56, 178, 580, 1922, 6466, 22022, 75788, 263152, 920768, 3243414, 11492460, 40934616, 146484296, 526389182, 1898722242, 6872300848, 24951521464, 90851221740, 331666951116, 1213729811070, 4451547793956
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=1..n} (1 + x^(k-1)*A(x))/(1-x^k) due to the q-binomial theorem.
a(n) ~ c * d^n / n^(3/2), where d = 3.881937422067584825536867239508405299121... and c = 4.5308041082663146457769... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.25760332825442180041464062514057254352... and A(r) = 5.79064730997128469298918813333150154669... satisfy A(r) = 1 / Sum_{n>=1} r^n/(1 + r^n*A(r)) and A(r) = Product_{n>=1} (1 + r^n*A(r))/(1-r^n). Note that r = 1/d and A(r) = s as given in the Mathematica program by Vaclav Kotesovec. - Paul D. Hanna, Mar 04 2024
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 178*x^5 + 580*x^6 + ...
such that the g.f. satisfies the identity:
A(x) = (1+x*A(x))/(1-x) * (1+x^2*A(x))/(1-x^2) * (1+x^3*A(x))/(1-x^3) * ...
A(x) = 1 + x*(1+A(x))/(1-x) + x^2*(1+A(x))*(1+x*A(x))/((1-x)*(1-x^2)) + x^3*(1+A(x))*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)*(1-x^3)) + ...
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MATHEMATICA
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nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k*A[x])/(1-x^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 04 2023 *)
(* Calculation of constants {d, c}: *) {1/r, Sqrt[-(s*(1 + s)* Log[r]*(s*(1 + s)*QPochhammer[r]^2*(Log[1 - r] + QPolyGamma[0, 1, r]) - r*Log[r]*QPochhammer[-s, r] * Derivative[0, 1][QPochhammer][r, r] + r*Log[r]*QPochhammer[r] * Derivative[0, 1][QPochhammer][-s, r]))/(2*Pi * QPochhammer[r]^2 * (s*Log[r]^2 + (1 + s)^2 * QPolyGamma[1, Log[-s]/Log[r], r]))]} /. FindRoot[{s*(1 + s)*QPochhammer[r] == QPochhammer[-s, r], Log[1 - r] + (1 + 2*s)*Log[r]/(1 + s) + QPolyGamma[0, Log[-s]/Log[r], r] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)
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PROG
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(PARI) {a(n) = my(A=1+x); for(i=1, n, A = prod(m=1, n, (1 + x^m*A)/(1 - x^m +x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + sum(m=1, n, x^m*prod(k=1, m, (1 + x^(k-1)*A)/(1 - x^k +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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