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A206301 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} A(x^k).

%I A206301 #10 Feb 20 2025 14:04:05
%S A206301 1,1,2,4,9,19,43,93,207,453,1003,2200,4860,10681,23552,51819,114186,
%T A206301 251326,553634,1218857,2684461,5910729,13016952,28662693,63120135,
%U A206301 138991543,306076520,673995311,1484205869,3268315926,7197126602,15848588048,34899932674
%N A206301 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} A(x^k).
%F A206301 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1+x*A(x) - x*A(x^2)/(1+x*A(x^2) - x*A(x^3)/(1+x*A(x^3) -...)))), a recursive continued fraction.
%e A206301 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 19*x^5 + 43*x^6 + 93*x^7 +...
%e A206301 such that, by definition,
%e A206301 A(x) = 1 + x*A(x) + x^2*A(x)*A(x^2) + x^3*A(x)*A(x^2)*A(x^3) + x^4*A(x)*A(x^2)*A(x^3)*A(x^4) + x^5*A(x)*A(x^2)*A(x^3)*A(x^4)*A(x^5) +...
%e A206301 The coefficients in Product_{k=1..n} A(x^k) begin:
%e A206301 n=2: [1, 1, 3, 5, 13, 25, 60, 124, 285, 609, 1369, 2970, 6611, ...];
%e A206301 n=3: [1, 1, 3, 6, 14, 28, 67, 139, 316, 683, 1523, 3317, 7369, ...];
%e A206301 n=4: [1, 1, 3, 6, 15, 29, 70, 145, 332, 713, 1596, 3468, 7717, ...];
%e A206301 n=5: [1, 1, 3, 6, 15, 30, 71, 148, 338, 728, 1627, 3540, 7868, ...];
%e A206301 n=6: [1, 1, 3, 6, 15, 30, 72, 149, 341, 734, 1642, 3570, 7941, ...];
%e A206301 n=7: [1, 1, 3, 6, 15, 30, 72, 150, 342, 737, 1648, 3585, 7971, ...]; ...
%o A206301 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*prod(k=1, m, subst(A, x, x^k +x*O(x^n))))); polcoeff(A, n)}
%o A206301 for(n=0, 35, print1(a(n), ", "))
%Y A206301 Cf. A206302, A091865.
%K A206301 nonn
%O A206301 0,3
%A A206301 _Paul D. Hanna_, Feb 06 2012