A182968 G.f.: A(x) = Product_{n>=1} 1/(1 - G_n(x)^n) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x): G_n(x) = G_{n-1}(G(x)) with G_0(x)=x.

1, 1, 3, 9, 31, 121, 540, 2692, 14938, 91599, 613393, 4443653, 34574511, 287092372, 2529894464, 23546940208, 230556268991, 2366688840376, 25394369705598, 284078044017592, 3305577581826931, 39928510807691616, 499751895103778173

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: A(x) = exp( Sum_{n>=1} Sum_{d|n} d*G_d(x)^n/n ) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 121*x^5 + 540*x^6 +...
Let G_n(x) denote the n-th iteration of G(x) = x + x^2, then
the logarithm of A(x) begins:
log(A(x)) = G(x) + [G(x)^2 + 2*G_2(x)^2]/2 + [G(x)^3 + 3*G_3(x)^3]/3 + [G(x)^4 + 2*G_2(x)^4 + 4*G_4(x)^4]/4 + [G(x)^5 + 5*G_5(x)^5]/5 +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 81*x^4/4 + 391*x^5/5 + 2159*x^6/6 + 13049*x^7/7 + 86257*x^8/8 + 618976*x^9/9 + 4763325*x^10/10 +...
The initial iterations of G(x) = x + x^2 begin:
G(G(x)) = x + 2*x^2 + 2*x^3 + x^4;
G_3(x) = x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8;
G_4(x) = x + 4*x^2 + 12*x^3 + 30*x^4 + 64*x^5 + 118*x^6 +...;
G_5(x) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + 630*x^6 +...;
G_6(x) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + 2170*x^6 +...;
See A122888 for a table of coefficients in iterations of x + x^2.
The g.f. equals the product:
A(x) = 1/[(1-x-x^2)*(1-(x+2*x^2+2*x^3+x^4)^2))*(1-(x+3*x^2+6*x^3+9*x^4+10*x^5+8*x^6+4*x^7+x^8)^3)...*(1-G_n(x)^n)*...]
where G_n(x) equals the n-th iteration of x+x^2.

Prog

(PARI) /* n-th Iteration of a function: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* G.f.: */
{a(n)=local(F); F=exp(sum(m=1, n+1, sumdiv(m, d, d*ITERATE(d, x+x^2, n)^m/m))); polcoeff(F, n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-ITERATE(k, x+x^2, n)^k), n)}

Crossrefs

Cf. A182970, A122888.

Sequence in context: A087648 A086616 A040027 * A071603 A349970 A090595

Adjacent sequences: A182965 A182966 A182967 * A182969 A182970 A182971

Keyword

nonn

Author

Paul D. Hanna, Dec 18 2010

Extensions

Name changed by Paul D. Hanna, Dec 19 2010

Status

approved