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

%I #5 Mar 30 2012 18:37:37

%S 1,1,3,6,14,31,72,166,390,922,2197,5273,12728,30892,75327,184476,

%T 453505,1118798,2768843,6872437,17103411,42670102,106697009,267359854,

%U 671260241,1688411587,4254084396,10735614274,27132998096,68671994940,174035109012,441607820562

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

%F 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.

%e 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 where the g.f. satisfies the identity:

%e 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 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)) +...

%o (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 (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 for(n=0,35,print1(a(n),", "))

%Y Cf. A190861, A190862.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 06 2012