%I #17 Dec 25 2023 17:56:18

%S 1,6,36,201,1116,6211,34581,192501,1071546,5964820,33203659,184830438,

%T 1028870637,5727277021,31881272165,177469235044,987894361908,

%U 5499180045361,30611553610680,170401260906615,948550017451201,5280167123920333,29392403504900866,163614780272069160,910772619152263675,5069876709305861450,28221807844318492913

%N Number of words of length n over {0,1,2,3,4,5} which have no factor iji with i>j.

%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. See Th. 3.12.

%F The g.f. can be written as either

%F 1/(1-x*(1+1/(1+x^2)+1/(1+2*x^2)+1/(1+3*x^2)+1/(1+4*x^2)+1/(1+5*x^2)))

%F or

%F -(1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)*(1+5*x^2)/(-1-15*x^2-85*x^4-225*x^6-274*x^8-120*x^10+6*x+120*x^11+548*x^9+675*x^7+340*x^5+75*x^3)

%p F312:=proc(k) local j,t1;

%p t1:=add(1/(1+j*x^2),j=0..k-1);

%p 1/(1-x*t1);

%p end;

%p seriestolist(series(F312(6),x,40));

%t CoefficientList[1/(1 - x*Sum[1/(1 + j*x^2), {j, 0, 5}]) + O[x]^30, x] (* _Jean-François Alcover_, Nov 24 2017 *)

%Y Cf. A005251, A123892, A123893, A123894.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, May 21 2013