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
Keywords
Examples
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_2(x) = 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) = Product_{n>=1} 1/(1-G_n(x)^n) = 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)*...)
where G_n(x) equals the n-th iteration of x+x^2.
Programs
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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)}
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).
Extensions
Name changed by Paul D. Hanna, Dec 19 2010