A180181 Triangle read by rows: T(n,k) is the number of compositions of n without 6's and having k parts; 1 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 0, 5, 10, 10, 5, 1, 1, 4, 15, 20, 15, 6, 1, 1, 5, 18, 35, 35, 21, 7, 1, 1, 6, 22, 52, 70, 56, 28, 8, 1, 1, 7, 27, 72, 121, 126, 84, 36, 9, 1, 1, 8, 33, 96, 190, 246, 210, 120, 45, 10, 1, 1, 10, 40, 125, 280, 432, 455, 330, 165, 55, 11, 1

Offset: 1

Author

Emeric Deutsch, Aug 15 2010

Keywords

• nonn
• tabl

Examples

			T(8,2)=5 because we have (1,7), (7,1), (3,5), (5,3), and (4,4).
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  6,  4,  1;
  0,  5, 10, 10,  5,  1;
  1,  4, 15, 20, 15,  6,  1;
		

References

• P. Chinn and S. Heubach, Compositions of n with no occurrence of k, Congressus Numerantium, 164 (2003), pp. 33-51 (see Table 8).
• R.P. Grimaldi, Compositions without the summand 1, Congressus Numerantium, 152, 2001, 33-43.

Links

• P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

Cf. A011973, A180177, A180178, A180179, A180180, A180182, A180183.

Programs

Maple

p := 6: T := proc (n, k) options operator, arrow: sum((-1)^(k-j)*binomial(k, j)*binomial(n-p*k+p*j-1, j-1), j = (p*k-n)/(p-1) .. k) end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
p := 6: g := z/(1-z)-z^p: G := t*g/(1-t*g): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 13 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form
with(combinat): m := 6: T := proc (n, k) local ct, i: ct := 0: for i to numbcomp(n, k) do if member(m, composition(n, k)[i]) = false then ct := ct+1 else end if end do: ct end proc: for n to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

Mathematica

p = 6; max = 14; g = z/(1-z) - z^p; G = t*g/(1-t*g); Gser = Series[G, {z, 0, max+1}]; t[n_, k_] := SeriesCoefficient[Gser, {z, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 28 2014, after Maple *)

Formula

Number of compositions of n without p's and having k parts = Sum_{j=(pk-n)/(p-1)..k} (-1)^(k-j)*binomial(k,j)*binomial(n-pk+pj-1, j-1).

For a given p, the g.f. of the number of compositions without p's is G(t,z) = t*g(z)/(1-t*g(z)), where g(z) = z/(1-z) - z^p; here z marks sum of parts and t marks number of parts.