A180179 Triangle read by rows: T(n,k) is the number of compositions of n without 4's and having k parts; 1 <= k <= n.

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 1, 2, 6, 4, 1, 1, 3, 7, 10, 5, 1, 1, 4, 9, 16, 15, 6, 1, 1, 6, 12, 23, 30, 21, 7, 1, 1, 6, 19, 32, 50, 50, 28, 8, 1, 1, 7, 24, 50, 76, 96, 77, 36, 9, 1, 1, 8, 30, 72, 120, 162, 168, 112, 45, 10, 1, 1, 9, 36, 99, 185, 267, 315, 274, 156, 55, 11, 1, 1, 10, 45

Offset

1,5

References

P. Chinn and S. Heubach, Compositions of n with no occurrence of k, Congressus Numerantium, 164 (2003), pp. 33-51 (see Table 6).

R.P. Grimaldi, Compositions without the summand 1, Congressus Numerantium, 152, 2001, 33-43.

Links

Table of n, a(n) for n=1..81.

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

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.

Example

T(7,3)=9 because we have (5,1,1), (1,5,1), (1,1,5), (3,3,1), (3,1,3), (1,3,3), (3,2,2), (2,3,2), and (2,2,3).
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  0,  3,  3,  1;
  1,  2,  6,  4,  1;
  1,  3,  7, 10,  5,  1;

Maple

p := 4: 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 := 4: 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 := 4: 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 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

Mathematica

p = 4; 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 *)

Crossrefs

Cf. A011973, A180177, A180178, A180180, A180181, A180182, A180183.

Sequence in context: A233530 A079123 A121548 * A346038 A215062 A215063

Adjacent sequences: A180176 A180177 A180178 * A180180 A180181 A180182

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 15 2010

Status

approved