|
%I #27 Feb 29 2024 15:53:54
%S 1,4,16,58,208,750,2708,9772,35256,127210,459012,1656228,5976040,
%T 21562946,77804232,280736004,1012961416,3655002994,13188110940,
%U 47585806908,171700784680,619536821778,2235434596432,8065973894524,29103931264328,105013830473538
%N Expansion of g.f.: (1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3 +11*x^4-22*x^5+6*x^6-6*x^7).
%C Number of words of length n over (0,1,2,3} which have no factor iji with i>j. - _N. J. A. Sloane_, May 21 2013
%H G. C. Greubel, <a href="/A123893/b123893.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, Annals. Combin., 7 (2003), 1-14.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,18,-11,22,-6,6).
%F a(0)=1, a(1)=4, a(2)=16, a(3)=58, a(4)=208, a(5)=750, a(6)=2708, a(n)= 4*a(n-1) -6*a(n-2) +18*a(n-3) -11*a(n-4) +22*a(n-5) -6*a(n-6) +6*a(n-7). - _Harvey P. Dale_, May 20 2012
%F G.f. can be written 1/(1-x*(1+1/(1+x^2)+1/(1+2*x^2)+1/(1+3*x^2))) which looks more symmetrical. _N. J. A. Sloane_, May 21 2013
%p seq(coeff(series((1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3+11*x^4 -22*x^5+6*x^6-6*x^7), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 06 2019
%t CoefficientList[Series[-(1+x^2) (1+2 x^2) (1+3 x^2)/(-1-6 x^2-11 x^4-6 x^6+4 x+18 x^3+22 x^5+6 x^7),{x,0,40}],x] (* or *) LinearRecurrence[ {4,-6,18,-11,22,-6,6},{1,4,16,58,208,750,2708},40] (* _Harvey P. Dale_, May 20 2012 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2 -18*x^3+11*x^4-22*x^5+6*x^6-6*x^7)) \\ _G. C. Greubel_, Aug 06 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3+11*x^4-22*x^5+6*x^6-6*x^7) )); // _G. C. Greubel_, Aug 06 2019
%o (Sage) ((1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3+11*x^4-22*x^5 +6*x^6-6*x^7)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 06 2019
%o (GAP) a:=[1,4,16,58,208,750,2708];; for n in [8..30] do a[n]:=4*a[n-1] -6*a[n-2]+18*a[n-3]-11*a[n-4]+22*a[n-5]-6*a[n-6]+6*a[n-7]; od; a; # _G. C. Greubel_, Aug 06 2019
%Y Cf. A005251, A123892, A123894, A225685.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 20 2006
|
