OFFSET
1,2
FORMULA
a(n) = [x^n] F_{n+1}(x) where F_{n+1}(x) = F_n(x+x^2) with F_1(x) = x+x^2 and F_0(x)=x for n>=1.
EXAMPLE
The first few iterations of (x+x^2) begin:
F(x) = x + x^2;
F(F(x)) = (1)*x + 2*x^2 + 2*x^3 + x^4;
F(F(F(x))) = x + (3)*x^2 + 6*x^3 + 9*x^4 + 10*x^5 +...;
F(F(F(F(x)))) = x + 4*x^2 + (12)*x^3 + 30*x^4 + 64*x^5 +...;
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + (70)*x^4 + 220*x^5 +...;
F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + (560)*x^5 +...;
coefficients enclosed in parenthesis form the initial terms of this sequence.
PROG
(PARI) {a(n)=local(F=x+x^2, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n, x)))}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 06 2005
STATUS
approved