Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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