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A190861 G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x))/(1-x^n).

%I A190861 #20 Mar 04 2024 11:51:43
%S A190861 1,2,6,18,56,178,580,1922,6466,22022,75788,263152,920768,3243414,
%T A190861 11492460,40934616,146484296,526389182,1898722242,6872300848,
%U A190861 24951521464,90851221740,331666951116,1213729811070,4451547793956
%N A190861 G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x))/(1-x^n).
%F A190861 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.
%F A190861 a(n) ~ c * d^n / n^(3/2), where d = 3.881937422067584825536867239508405299121... and c = 4.5308041082663146457769... - _Vaclav Kotesovec_, Oct 04 2023
%F A190861 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
%e A190861 G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 178*x^5 + 580*x^6 + ...
%e A190861 such that the g.f. satisfies the identity:
%e A190861 A(x) = (1+x*A(x))/(1-x) * (1+x^2*A(x))/(1-x^2) * (1+x^3*A(x))/(1-x^3) * ...
%e A190861 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)) + ...
%t A190861 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 *)
%t A190861 (* 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 *)
%o A190861 (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)}
%o A190861 (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)}
%Y A190861 Cf. A209357, A190862.
%K A190861 nonn
%O A190861 0,2
%A A190861 _Paul D. Hanna_, May 21 2011