A097191 G.f. A(x) has the property that the first (n+1) terms of A(x)^(n+1) form the n-th row polynomial R_n(y) of triangle A097190 and satisfy R_n(1/3) = 9^n for all n>=0.

1, 12, 60, 90, -558, -2916, 2160, 61155, 137619, -767880, -4940676, 0, 95128668, 285386004, -974126979, -8413235910, -6504831279, 142312459626, 552074177142, -1081032363522, -13861905214518, -20792857821777, 204246531941697, 1012677253935633, -890531709052761

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• sign

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A097190, A097193.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 27*x/(1-(1-3*x)^9) )); // G. C. Greubel, Sep 17 2019

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 *)

PARI

a(n)=polcoeff(27*x/(1-(1-3*x)^9)+x*O(x^n),n,x)

Sage

def A097191_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(27*x/(1-(1-3*x)^9)).list()
A097191_list(30) # G. C. Greubel, Sep 17 2019

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.