A180180 Triangle read by rows: T(n,k) is the number of compositions of n without 5's and having k parts; 1 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 0, 4, 6, 4, 1, 1, 3, 10, 10, 5, 1, 1, 4, 12, 20, 15, 6, 1, 1, 5, 15, 31, 35, 21, 7, 1, 1, 6, 19, 44, 65, 56, 28, 8, 1, 1, 8, 24, 60, 106, 120, 84, 36, 9, 1, 1, 8, 33, 80, 160, 222, 203, 120, 45, 10, 1, 1, 9, 40, 111, 230, 372, 420, 322, 165, 55, 11, 1

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 7).

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

Links

Table of n, a(n) for n=1..78.

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

Maple

p:= 5: 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:=5: g:=z/(1-z)-z^(p): G:=t*g/(1-t*g): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n)); # yields sequence in triangular form
with(combinat): m := 5: 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 = 5; 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, A180179, A180181, A180182, A180183.

Sequence in context: A216210 A186332 A129571 * A034931 A248473 A307116

Adjacent sequences: A180177 A180178 A180179 * A180181 A180182 A180183

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 15 2010

Status

approved