A112319 Coefficients of x^n in the (n-1)-th iteration of (x + x^2) for n>=1.

1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, 1059312264, 27715541568, 800423573676, 25289923553700, 867723362137464, 32128443862364255, 1276818947065793736, 54208515369076658640, 2448636361058495090816, 117254071399557173396416

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

a(n) = [x^n] F_{n-1}(x) where F_n(x) = F_{n-1}(x+x^2) with F_1(x) = x+x^2 and F_0(x)=x for n>=1.

Example

The iterations of (x+x^2) begin:
F(x) = x + (1)*x^2
F(F(x)) = x + 2*x^2 + (2)*x^3 + x^4
F(F(F(x))) = x + 3*x^2 + 6*x^3+ (9)*x^4 +...
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 + (630)*x^6 +...
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, 30, print1(a(n), ", "))

Crossrefs

Cf. A112317, A112320.

Sequence in context: A152917 A213236 A000169 * A232552 A038038 A368790

Adjacent sequences: A112316 A112317 A112318 * A112320 A112321 A112322

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2005

Status

approved