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%I #23 Mar 04 2024 09:05:57
%S 1,1,2,4,7,14,27,53,104,208,415,836,1690,3434,7004,14342,29460,60707,
%T 125443,259883,539689,1123226,2342493,4894590,10245321,21481047,
%U 45108768,94863801,199772929,421245065,889331420,1879723964,3977402460,8424718846
%N G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n) * (1 + x^(2n)*A(x)).
%F 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.
%F From _Vaclav Kotesovec_, Mar 03 2024: (Start)
%F Let A(x) = y, then 2*y*(1 + y) = QPochhammer(-1, x) * QPochhammer(-y, x^2).
%F a(n) ~ c * d^n / n^(3/2), where d = 2.20229791253644493239805950840417681972879718454582447550768622636671... and c = 9.92694112477002167508700773789825154871250555780774205172995613775...
%F Radius of convergence:
%F r = 1/d = 0.45407117461609608946909851977877786178200148047136427660297778018...
%F A(r) = s = 8.6584215712749049134273598177515922912152713325328273868580739614...
%F (End)
%F 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
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 + ...
%e 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)) * ...
%e 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)) + ...
%t 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 *)
%t (* 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 *)
%o (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)}
%o (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)}
%Y Cf. A145267.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 21 2011
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