OFFSET
1,2
FORMULA
G.f.: A(x) = x+x^2 o x+x^3 o...o x+x^n o...o (x) o...o x+x^n o...o x+x^3 o x+x^2.
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 13*x^4 + 39*x^5 + 113*x^6 +...
G.f. A(x) equals the limit of symmetric compositions starting with:
(1) x+x^2 o x+x^2 = x + 2*x^2 + 2*x^3 + x^4 ;
(2) x+x^2 o x+x^3 o x+x^3 o x+x^2 = x + 2*x^2 + 4*x^3 + 11*x^4 +...;
(3) x+x^2 o x+x^3 o x+x^4 o x+x^4 o x+x^3 o x+x^2 = x + 2*x^2 + 4*x^3 + 13*x^4 +...
PROG
(PARI) {a(n)=local(F=x); if(n<1, 0, for(k=2, n, F=subst(subst(x+x^(n-k+2), x, F), x, x+x^(n-k+2)+x*O(x^n)); ); return(polcoeff(F, n)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 16 2009
STATUS
approved