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%I #15 Sep 08 2022 08:45:31
%S 0,0,0,1,6,24,81,252,756,2241,6642,19764,59049,176904,530712,1592865,
%T 4780782,14346720,43046721,129146724,387440172,1162300833,3486843450,
%U 10460412252,31381059609,94143001680,282429005040,847287546561
%N Inverse binomial transform of (A113405 preceded by 0).
%H G. C. Greubel, <a href="/A133474/b133474.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,9).
%F 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)?
%F From _R. J. Mathar_, Apr 02 2008: (Start)
%F O.g.f.: x^3/((1-3*x)*(1-3*x+3*x^2)).
%F a(n) = 6*a(n-1) - 12*a(n-2) + 9*a(n-3). (End)
%p 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
%t LinearRecurrence[{6,-12,9}, {0,0,0,1}, 30] (* _G. C. Greubel_, Nov 21 2019 *)
%o (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
%o (Magma) R<x>:=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
%o (Sage)
%o def A133474_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(x^3/((1-3*x)*(1-3*x+3*x^2))).list()
%o A133474_list(30) # _G. C. Greubel_, Nov 21 2019
%o (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
%Y Cf. A047790, A131571.
%K nonn,easy
%O 0,5
%A _Paul Curtz_, Nov 29 2007
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