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proposed

(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)))}

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Coefficients Coefficient of x^n in the (n+1)-th iteration of (x + x^2) for n>=1.

{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)))}

(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), ", "))

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Coefficients of x^n in the (n+1)-th self-composition iteration of (x + x^2) for n>=1.

Initial terms in self-compositions The first few iterations of (x+x^2) arebegin:

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.

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editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Sep 06 2005

nonn,new

nonn

Paul D . Hanna (pauldhanna(AT)juno.com), Sep 06 2005

Coefficients of x^n in the (n+1)-th self-composition of (x + x^2) for n>=1.

1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, 10052947476, 261595087182, 7509722346204, 235808741944100, 8040824716606176, 295914258931377276, 11690732617035570008, 493527339623630078552

1,2

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.

Initial terms in self-compositions of (x+x^2) are:

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 +...

{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)))}

Cf. A112317, A112319.

nonn

Paul D Hanna (pauldhanna(AT)juno.com), Sep 06 2005

approved