A123888 - OEIS

%I #17 Oct 29 2023 12:36:10

%S 1,3,9,24,63,163,420,1080,2775,7128,18307,47016,120744,310086,796338,

%T 2045089,5252025,13487805,34638234,88954965,228446569,586677030,

%U 1506654000,3869260527,9936705456,25518600937,65534698260,168300632412,432215354952,1109978675532

%N Expansion of g.f.: x/((1-x^2)^3 -1+x).

%H G. C. Greubel, <a href="/A123888/b123888.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Burstein and T. Mansour, <a href="https://arxiv.org/abs/math/0112281">Words restricted by 3-letter ...</a>, arXiv:math/0112281 [math.CO], 2001.

%H A. Burstein and T. Mansour, <a href="https://doi.org/10.1007/s000260300000">Words Restricted by 3-Letter Generalized Multipermutation Patterns</a>, Annals. Combin., 7 (2003), 1-14.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,0,1).

%p seq(coeff(series(1/(1-3*x+3*x^3-x^5), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 07 2019

%t CoefficientList[Series[x/((1-x^2)^3 -1+x), {x,0,30}], x] (* _G. C. Greubel_, Aug 07 2019 *)

%t LinearRecurrence[{3,0,-3,0,1},{1,3,9,24,63},30] (* _Harvey P. Dale_, Oct 29 2023 *)

%o (PARI) my(x='x+O('x^30)); Vec(x/((1-x^2)^3 -1+x)) \\ _G. C. Greubel_, Aug 07 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x/((1-x^2)^3 -1+x) )); // _G. C. Greubel_, Aug 07 2019

%o (Sage)

%o def A123880_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x/((1-x^2)^3 -1+x) ).list()

%o A123888_list(30) # _G. C. Greubel_, Aug 07 2019

%o (GAP) a:=[1,3,9,24,63];; for n in [6..30] do a[n]:=3*a[n-1]-3*a[n-3]+ a[n-5]; od; a; # _G. C. Greubel_, Aug 07 2019

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 20 2006