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%I #24 Sep 08 2022 08:45:28
%S 1,5,25,115,525,2405,11025,50525,231525,1060979,4862065,22280875,
%T 102104045,467901065,2144199765,9825991725,45028504525,206347239279,
%U 945605099381,4333321849875,19857843661645,91000384646845,417017585036185,1911021221349165
%N Expansion of g.f.: (1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1-5*x+10*x^2 -40*x^3+35*x^4-105*x^5 +50*x^6-100*x^7+24*x^8-24*x^9).
%C Number of words of length n over (0,1,2,3,4} which have no factor iji with i>j. - _N. J. A. Sloane_, May 21 2013
%H G. C. Greubel, <a href="/A123894/b123894.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 generalized multipermutation patterns</a>, arXiv:math/0112281 [math.CO], 2001; Annals. Combin., 7 (2003), 1-14.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,40,-35,105,-50,100,-24,24).
%F G.f. may be written more symmetrically as 1/(1-x*(1 +1/(1+x^2) +1/(1+2*x^2) +1/(1+3*x^2) +1/(1+4*x^2))). - _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^2)/(1 -5*x +10*x^2 -40*x^3+35*x^4-105*x^5+50*x^6-100*x^7+24*x^8-24*x^9), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 06 2019
%t CoefficientList[1/(1 - x(1 + 1/(1+x^2) + 1/(1+2x^2) + 1/(1+3x^2) + 1/(1+4x^2))) + O[x]^30, x] (* _Jean-François Alcover_, Jan 09 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1 -5*x +10*x^2 -40*x^3+35*x^4-105*x^5+50*x^6-100*x^7+24*x^8-24*x^9)) \\ _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^2)/(1-5*x+10*x^2 -40*x^3+35*x^4-105*x^5 +50*x^6-100*x^7+24*x^8-24*x^9) )); // _G. C. Greubel_, Aug 06 2019
%o (Sage) ((1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1-5*x+10*x^2-40*x^3 +35*x^4-105*x^5 +50*x^6-100*x^7+24*x^8-24*x^9)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 06 2019
%o (GAP) a:=[1, 5, 25, 115, 525, 2405, 11025, 50525, 231525];; for n in [10..30] do a[n]:=5*a[n-1]-10*a[n-2] +40*a[n-3]-35*a[n-4]+105*a[n-5] -50*a[n-6]+100*a[n-7]-24*a[n-8]+24*a[n-9]; od; a; # _G. C. Greubel_, Aug 06 2019
%Y Cf. A005251, A123892, A123893, A225685.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 20 2006
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