OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = 27*x/(1-(1-3*x)^9).
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A097193.
a(18*n+11) = 0 for n>=0.
EXAMPLE
A(x) = 1 + 12*x + 60*x^2 + 90*x^3 - 558*x^4 - 2916*x^5 + 2160*x^6 +...
For n>=0, the first (n+1) coefficients of A(x)^(n+1) forms the
n-th row polynomial R_n(y) of triangle A097190:
A^1 = {1, _12, 60, 90, -558, -2916, 2160, ...}
A^2 = {1, 24, _264, 1620, 4644, -8424, -124524, ...}
A^3 = {1, 36, 612, _6318, 41526, 151956, -16308, ...}
A^4 = {1, 48, 1104, 15912, _156744, 1061424, 4423032, ...}
A^5 = {1, 60, 1740, 32130, 417690, _3966732, 27243000, ...}
A^6 = {1, 72, 2520, 56700, 912492, 11027016, _101653164, ...}
These row polynomials satisfy: R_n(1/3) = 9^n:
9^1 = 1 + 24/3;
9^2 = 1 + 36/3 + 612/3^2;
9^3 = 1 + 48/3 + 1104/3^2 + 15912/3^3;
9^4 = 1 + 60/3 + 1740/3^2 + 32130/3^3 + 417690/3^4.
MAPLE
seq(coeff(series(27*x/(1-(1-3*x)^9), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019
MATHEMATICA
CoefficientList[Series[27*x/(1-(1-3*x)^9), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2019 *)
PROG
(PARI) a(n)=polcoeff(27*x/(1-(1-3*x)^9)+x*O(x^n), n, x)
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 27*x/(1-(1-3*x)^9) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A097191_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P(27*x/(1-(1-3*x)^9)).list()
A097191_list(30) # G. C. Greubel, Sep 17 2019
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Aug 03 2004
STATUS
approved