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A209357 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^(n+1)*A(x)) / (1 - x^n).

%I A209357 #15 Jun 10 2025 10:34:20
%S A209357 1,1,3,6,14,31,72,166,390,922,2197,5273,12728,30892,75327,184476,
%T A209357 453505,1118798,2768843,6872437,17103411,42670102,106697009,267359854,
%U A209357 671260241,1688411587,4254084396,10735614274,27132998096,68671994940,174035109012,441607820562
%N A209357 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^(n+1)*A(x)) / (1 - x^n).
%H A209357 Vaclav Kotesovec, <a href="/A209357/b209357.txt">Table of n, a(n) for n = 0..455</a>
%F A209357 G.f. satisfies: A(x) = Sum_{n>=0} x^n*Product_{k=1..n} (1 + x^k*A(x))/(1-x^k) due to the q-binomial theorem.
%F A209357 a(n) ~ c * d^n / n^(3/2), where d = 2.6481816651621274063587047915... and c = 7.257947883786923940523402074... - _Vaclav Kotesovec_, Jun 10 2025
%e A209357 G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 72*x^6 + 166*x^7 +...
%e A209357 where the g.f. satisfies the identity:
%e A209357 A(x) = (1+x^2*A(x))/(1-x) * (1+x^3*A(x))/(1-x^2) * (1+x^4*A(x))/(1-x^3) *...
%e A209357 A(x) = 1 + x*(1+x*A(x))/(1-x) + x^2*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^3*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) +...
%t A209357 (* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(s*(1 + r*s)*Log[r]*(s*(1 + r*s)*(-QPochhammer[r]*(Log[1 - r] + Log[r] + QPolyGamma[0, 1, r]) + r*Log[r]*Derivative[0, 1][QPochhammer][r, r]) - r*Log[r]*Derivative[0, 1][QPochhammer][-r*s, r])) / (2*Pi*QPochhammer[r] * (r*s*Log[r]^2 + (1 + r*s)^2*QPolyGamma[1, Log[-r*s]/Log[r], r]))]} /. FindRoot[{s*(1 + r*s) == QPochhammer[-r*s, r]/QPochhammer[r], Log[1-r] + r*s*Log[r]/(1 + r*s) + QPolyGamma[0, Log[-r*s]/Log[r], r] == -Log[r]}, {r, 2/5}, {s, 2}, WorkingPrecision -> 120]] (* _Vaclav Kotesovec_, Jun 10 2025 *)
%o A209357 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+x^(m+1)*A)/(1-x^m+x*O(x^n)))); polcoeff(A, n)}
%o A209357 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*prod(k=1, m, (1+x^k*A)/(1-x^k+x*O(x^n))))); polcoeff(A, n)}
%o A209357 for(n=0,35,print1(a(n),", "))
%Y A209357 Cf. A190861, A190862.
%K A209357 nonn
%O A209357 0,3
%A A209357 _Paul D. Hanna_, Mar 06 2012