A225682 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.

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

Offset

0,5

Comments

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.

For example, G_2, G_3, G_4, G_5, G_6 are g.f.'s for A000079, A076264, A072335, A200781, A200782.

Links

Table of n, a(n) for n=0..90.

Example

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],
...

Maple

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)];

Mathematica

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 *)

Crossrefs

Cf. A000079, A076264, A072335, A200781, A200782.

Sequence in context: A096335 A191910 A129503 * A144185 A143987 A309013

Adjacent sequences: A225679 A225680 A225681 * A225683 A225684 A225685

Keyword

sign,tabl

Author

Murray R. Bremner and N. J. A. Sloane, May 17 2013

Status

approved