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A225682
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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.
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5
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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, 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, 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
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OFFSET
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0,5
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COMMENTS
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Corresponding to row n of this triangle, define a generating function G_n(x) = 1/(Sum_{k=0..n} T(n,k)*x^k).
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.
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LINKS
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EXAMPLE
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Triangle begins:
[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],
[1, -7, 0, 35, -35, 0, 7, -1],
[1, -8, 0, 56, -70, 0, 28, -8, 0],
...
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MAPLE
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f:=proc(n) local k, s;
s:=k->if k mod 3 = 0 then 1 elif k mod 3 = 1 then -1 else 0; fi;
[seq(s(k)*binomial(n, k), k=0..n)];
end;
[seq(f(n), n=0..12)];
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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