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A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1. 15
1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, 3479498880, 90230486346, 2584679465160, 81056989408928, 2762187020749144, 101633218030586364, 4015771398425994048, 169588657820702174728 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Forms a diagonal of the tables A122888 and A185755.

LINKS

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..300 (first 100 terms from Paul D. Hanna)

FORMULA

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

EXAMPLE

The initial iterations of x + x^2 begin:

F(x) = (1)*x + 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+ 10*x^5+ 8*x^6+ 4*x^7+ x^8;

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 + (2170)*x^6 +...;

where the terms in parenthesis illustrate how to form 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, G=subst(F, x, G)); return(polcoeff(G, n, x)))}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A112319, A112320, A122888, A185755, A135080, A166900.

Sequence in context: A205569 A108204 A088160 * A089459 A027882 A106209

Adjacent sequences:  A112314 A112315 A112316 * A112318 A112319 A112320

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 03 2005

EXTENSIONS

Added cross-references and comments; name and example changed by Paul D. Hanna, Feb 04 2011

STATUS

approved

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Last modified November 17 00:14 EST 2018. Contains 317275 sequences. (Running on oeis4.)