A133474 Inverse binomial transform of (A113405 preceded by 0).

0, 0, 0, 1, 6, 24, 81, 252, 756, 2241, 6642, 19764, 59049, 176904, 530712, 1592865, 4780782, 14346720, 43046721, 129146724, 387440172, 1162300833, 3486843450, 10460412252, 31381059609, 94143001680, 282429005040, 847287546561

Offset: 0

Author

Paul Curtz, Nov 29 2007

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A047790, A131571.

Programs

GAP

a:=[0,0,1];; for n in [4..30] do a[n]:=6*a[n-1]-12*a[n-2]+9*a[n-3]; od; a; # G. C. Greubel, Nov 21 2019

Magma

R:=PowerSeriesRing(Integers(), 30); [0,0,0] cat Coefficients(R!( x^3/((1-3*x)*(1-3*x+3*x^2)) )); // G. C. Greubel, Nov 21 2019

Maple

seq(coeff(series(x^3/((1-3*x)(1-3*x+3*x^2)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Nov 21 2019

Mathematica

LinearRecurrence[{6,-12,9}, {0,0,0,1}, 30] (* G. C. Greubel, Nov 21 2019 *)

PARI

my(x='x+O('x^30)); concat([0,0,0], Vec(x^3/((1-3*x)*(1-3*x+3*x^2)))) \\ G. C. Greubel, Nov 21 2019

Sage

def A133474_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^3/((1-3*x)*(1-3*x+3*x^2))).list()
A133474_list(30) # G. C. Greubel, Nov 21 2019

Formula

b(n) = a(n) with one 0; c(n)=1, 3, 6, 9, 9, 0, -27, ... = A057083; b(n+1) = 3*b(n) + c(n)?

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

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

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