OFFSET
1,3
COMMENTS
All self-compositions of A(x) may be expressed as a finite sum involving powers of A(x) and x.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 11*x^4 + 88*x^5 + 888*x^6 + 10572*x^7 + 143214*x^8 + 2159154*x^9 + 35702442*x^10 + 640873656*x^11 + 12394383780*x^12 + 256762580460*x^13 + 5671209169168*x^14 + 133041670286160*x^15 +...
A(A(x)) = x + 2*x^2 + 6*x^3 + 33*x^4 + 264*x^5 + 2664*x^6 + 31716*x^7 + 429642*x^8 + 6477462*x^9 + 107107326*x^10 + 1922620968*x^11 + 37183151340*x^12 + 770287741380*x^13 + 17013627507504*x^14 + 399125010858480*x^15 +...
so that A(A(x)) + 2*x + x^2 = 3*A(x).
Self-compositions of A=A(x) may be expressed in terms of A and x:
A(A(x)) = 3*A - 2*x - x^2 ;
A(A(A(x))) = (7*A - A^2) - 6*x - 3*x^2 ;
A(A(A(A(x)))) = (15*A - 12*A^2) + (-14 + 12*A)*x +
(-11 + 6*A)*x^2 - 4*x^3 - x^4 ;
A(A(A(A(A(x))))) = (31*A - 83*A^2 + 14*A^3 - A^4) +
(-12*A^2 + 120*A - 30)*x + (-6*A^2 + 60*A - 63)*x^2 - 48*x^3 - 12*x^4 .
PROG
(PARI) {a(n)=local(A=x+x^2); if(n<1, 0, for(i=3, n+1, A=A+polcoeff(subst(A, x, A+x*O(x^i)), i)*x^i); polcoeff(A, n))}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 27 2008
STATUS
approved