OFFSET
1,2
COMMENTS
Limit a(n+1)/a(n) ~ 3.80825961708875...
FORMULA
G.f.: A(x) = x/(1-x) o x/(1-x^2) o x/(1-x^3) o...o (x) o...o x/(1-x^3) o x/(1-x^2) o x/(1-x).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 245*x^6 +...
A(x) is the limit of the compositions beginning in the following manner:
(1) x/(1-x) o x/(1-x) = x/(1-2*x);
(2) x/(1-x) o x/(1-x^2) o x/(1-x^2) o x/(1-x) = (x-3*x^2+2*x^3)/(1-5*x+6*x^2-x^4);
(3) x/(1-x) o x/(1-x^2) o x/(1-x^3) o x/(1-x^3) o x/(1-x^2) o x/(1-x); ...
PROG
(PARI) {a(n)=local(F=x); if(n<1, 0, for(k=1, n, F=subst(subst(x/(1-x^(n-k+1)), x, F), x, x/(1-x^(n-k+1) +x*O(x^n))); ); return(polcoeff(F, n)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 12 2009
STATUS
approved