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

1, 1, 2, 4, 9, 19, 43, 93, 207, 453, 1003, 2200, 4860, 10681, 23552, 51819, 114186, 251326, 553634, 1218857, 2684461, 5910729, 13016952, 28662693, 63120135, 138991543, 306076520, 673995311, 1484205869, 3268315926, 7197126602, 15848588048, 34899932674

Offset: 0

Author

Paul D. Hanna, Feb 06 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 19*x^5 + 43*x^6 + 93*x^7 +...
such that, by definition,
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) +...
The coefficients in Product_{k=1..n} A(x^k) begin:
n=2: [1, 1, 3, 5, 13, 25, 60, 124, 285, 609, 1369, 2970, 6611, ...];
n=3: [1, 1, 3, 6, 14, 28, 67, 139, 316, 683, 1523, 3317, 7369, ...];
n=4: [1, 1, 3, 6, 15, 29, 70, 145, 332, 713, 1596, 3468, 7717, ...];
n=5: [1, 1, 3, 6, 15, 30, 71, 148, 338, 728, 1627, 3540, 7868, ...];
n=6: [1, 1, 3, 6, 15, 30, 72, 149, 341, 734, 1642, 3570, 7941, ...];
n=7: [1, 1, 3, 6, 15, 30, 72, 150, 342, 737, 1648, 3585, 7971, ...]; ...
		

Crossrefs

Cf. A206302, A091865.

Programs

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)}
for(n=0, 35, print1(a(n), ", "))

Formula

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.