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%I #21 Jan 14 2014 10:03:39
%S 1,1,-1,1,-2,0,1,-3,0,1,1,-4,0,4,-1,1,-5,0,10,-5,0,1,-6,0,20,-15,0,1,
%T 1,-7,0,35,-35,0,7,-1,1,-8,0,56,-70,0,28,-8,0,1,-9,0,84,-126,0,84,-36,
%U 0,1,1,-10,0,120,-210,0,210,-120,0,10,-1,1,-11,0,165,-330,0,462,-330,0,55,-11,0,1,-12,0,220,-495,0,924,-792,0,220,-66,0,1
%N Triangle read by rows: T(n,k) (0 <= k <= n) = chi(k)*binomial(n,k), where chi(k) = 1,-1,0 according as k == 0,1,2 mod 3.
%C Corresponding to row n of this triangle, define a generating function G_n(x) = 1/(Sum_{k=0..n} T(n,k)*x^k).
%C Then G_n(x) is the g.f. for the number of words of length n over an alphabet of size n which do not contain any strictly decreasing factor (consecutive subword) of length 3.
%C For example, G_2, G_3, G_4, G_5, G_6 are g.f.'s for A000079, A076264, A072335, A200781, A200782.
%e Triangle begins:
%e [1],
%e [1, -1],
%e [1, -2, 0],
%e [1, -3, 0, 1],
%e [1, -4, 0, 4, -1],
%e [1, -5, 0, 10, -5, 0],
%e [1, -6, 0, 20, -15, 0, 1],
%e [1, -7, 0, 35, -35, 0, 7, -1],
%e [1, -8, 0, 56, -70, 0, 28, -8, 0],
%e ...
%p f:=proc(n) local k,s;
%p s:=k->if k mod 3 = 0 then 1 elif k mod 3 = 1 then -1 else 0; fi;
%p [seq(s(k)*binomial(n,k),k=0..n)];
%p end;
%p [seq(f(n),n=0..12)];
%t chi[k_] := Switch[Mod[k, 3], 0, 1, 1, -1, 2, 0]; t[n_, k_] := chi[k]*Binomial[n, k]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%Y Cf. A000079, A076264, A072335, A200781, A200782.
%K sign,tabl
%O 0,5
%A _Murray R. Bremner_ and _N. J. A. Sloane_, May 17 2013
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