A135254 Binomial transform of A131666.

0, 0, 1, 4, 12, 33, 90, 252, 729, 2160, 6480, 19521, 58806, 176904, 531441, 1595052, 4785156, 14353281, 43053282, 129146724, 387420489, 1162241784, 3486725352, 10460235105, 31380882462, 94143001680, 282429536481, 847289140884

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (6,-12,9).

Formula

From R. J. Mathar, Apr 02 2008: (Start)

O.g.f.: x^2*(1-2*x)/((1 - 3*x + 3*x^2)*(1-3*x)).

a(n) = 6*a(n-1) - 12*a(n-2) + 9*a(n-3). (End)

Maple

seq(coeff(series(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 21 2019

Mathematica

CoefficientList[Series[x^2(2x-1)/((3x^2-3x+1)(3x-1)), {x, 0, 30}], x] (* or *) LinearRecurrence[{6, -12, 9}, {0, 0, 1, 4}, 30] (* Harvey P. Dale, May 26 2011 *)

Prog

(PARI) my(x='x+O('x^30)); concat([0, 0], Vec(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)))) \\ G. C. Greubel, Nov 21 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0, 0] cat Coefficients(R!( x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)) )); // G. C. Greubel, Nov 21 2019
(Sage)
def A135254_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x))).list()
A135254_list(30) # G. C. Greubel, Nov 21 2019
(GAP) a:=[0, 1, 4];; for n in [4..30] do a[n]:=6*a[n-1]-12*a[n-2]+9*a[n-3]; od; Concatenation([0], a); # G. C. Greubel, Nov 21 2019

Crossrefs

Cf. A133474.

Sequence in context: A027941 A293064 A219092 * A326804 A000754 A317974

Adjacent sequences: A135251 A135252 A135253 * A135255 A135256 A135257

Keyword

nonn

Author

Paul Curtz, Nov 30 2007

Extensions

More terms from R. J. Mathar, Apr 02 2008

Status

approved