OFFSET
1,3
FORMULA
G.f.: A(x) = x*exp( Sum_{n>=1} Sum_{d|n} d*A_d(x)^n/n ) where A_n(x) denotes the n-th iteration of A(x).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 61*x^5 + 365*x^6 +...
Let A_n(x) denote the n-th iteration of g.f. A(x), then
the logarithm of A(x)/x begins:
log(A(x)/x) = A(x) + [A(x)^2 + 2*A_2(x)^2]/2 + [A(x)^3 + 3*A_3(x)^3]/3 + [A(x)^4 + 2*A_2(x)^4 + 4*A_4(x)^4]/4 + [A(x)^5 + 5*A_5(x)^5]/5 +...
Explicitly,
log(A(x)/x) = x + 5*x^2/2 + 28*x^3/3 + 189*x^4/4 + 1431*x^5/5 + 11858*x^6/6 + 105533*x^7/7 + 996541*x^8/8 + 9901306*x^9/9 + 102895485*x^10/10 +...
The initial iterations of A(x) begin:
A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 236*x^5 + 1571*x^6 +...
A_3(x) = x + 3*x^2 + 15*x^3 + 90*x^4 + 613*x^5 + 4586*x^6 +...
A_4(x) = x + 4*x^2 + 24*x^3 + 168*x^4 + 1304*x^5 + 10926*x^6 +...
A_5(x) = x + 5*x^2 + 35*x^3 + 280*x^4 + 2445*x^5 + 22775*x^6 +...
A_6(x) = x + 6*x^2 + 48*x^3 + 432*x^4 + 4196*x^5 + 43105*x^6 +...
A_7(x) = x + 7*x^2 + 63*x^3 + 630*x^4 + 6741*x^5 + 75796*x^6 +...
A_8(x) = x + 8*x^2 + 80*x^3 + 880*x^4 + 10288*x^5 + 125756*x^6 +...
The g.f. equals the product:
A(x) = x/[(1 - A(x))*(1 - A(A(x))^2))*(1 - A(A(A(x)))^3)*(1 - A(A(A(A(x))))^4)* ...*(1 - A_n(x)^n)*...]
where A_n(x) equals the n-th iteration of A(x).
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=x+x^2+x*O(x^n)); for(i=0, n, F=x*exp(sum(m=1, n+1, 1/m*sumdiv(m, d, d*ITERATE(d, F, n)^m)))); polcoeff(F, n)}
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x/prod(k=1, n, 1-ITERATE(k, A, n)^k)); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 18 2010
EXTENSIONS
Name changed by Paul D. Hanna, Dec 19 2010
STATUS
approved