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A190822
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G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n) * (1 + x^(2n)*A(x)).
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3
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1, 1, 2, 4, 7, 14, 27, 53, 104, 208, 415, 836, 1690, 3434, 7004, 14342, 29460, 60707, 125443, 259883, 539689, 1123226, 2342493, 4894590, 10245321, 21481047, 45108768, 94863801, 199772929, 421245065, 889331420, 1879723964, 3977402460, 8424718846
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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} x^(n*(n+1)/2) * Product_{k=1..n} (1 + x^k*A(x)) / (1 - x^k) due to a Lebesgue identity.
Let A(x) = y, then 2*y*(1 + y) = QPochhammer(-1, x) * QPochhammer(-y, x^2).
a(n) ~ c * d^n / n^(3/2), where d = 2.20229791253644493239805950840417681972879718454582447550768622636671... and c = 9.92694112477002167508700773789825154871250555780774205172995613775...
Radius of convergence:
r = 1/d = 0.45407117461609608946909851977877786178200148047136427660297778018...
A(r) = s = 8.6584215712749049134273598177515922912152713325328273868580739614...
(End)
The values r and A(r) given above also satisfy A(r) = 1 / Sum_{n>=1} r^(2*n)/(1 + r^(2*n)*A(r)). - Paul D. Hanna, Mar 03 2024
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 + ...
G.f.: A(x) = (1+x)*(1+x^2*A(x)) * (1+x^2)*(1+x^4*A(x)) * (1+x^3)*(1+x^6*A(x)) * ...
G.f.: A(x) = 1 + x*(1+x*A(x))/(1-x) + x^3*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^6*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) + ...
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MATHEMATICA
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nmax = 40; A[_] = 0; Do[A[x_] = Product[(1 + x^k)*(1 + x^(2*k)*A[x]), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Mar 03 2024 *)
(* Calculation of constants {d, c}: *) Chop[{1/r, Sqrt[(-r*s*(1 + s) * Log[r^2]^2 * (s*(1 + s)*Derivative[0, 1][QPochhammer][-1, r] + r*QPochhammer[-1, r]^2 * Derivative[0, 1][QPochhammer][-s, r^2]))/(2*Pi * QPochhammer[-1, r]* (s*Log[r^2]^2 + (1 + s)^2 * QPolyGamma[1, Log[-s]/Log[r^2], r^2]))]} /. FindRoot[{2*s*(1 + s) == QPochhammer[-1, r]*QPochhammer[-s, r^2], 1 + s/(1 + s) + (Log[1 - r^2] + QPolyGamma[0, Log[-s]/Log[r^2], r^2])/Log[r^2] == 0}, {r, 1/2}, {s, 8}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Mar 03 2024 *)
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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) * (1 + x^(2*m)*A+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + sum(m=1, sqrtint(2*n), x^(m*(m+1)/2) * prod(k=1, m, (1 + A*x^k)/(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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