A180177 Triangle read by rows: T(n,k) is the number of compositions of n without 2's and having k parts; 1<=k<=n.

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 3, 3, 4, 0, 1, 1, 4, 6, 4, 5, 0, 1, 1, 5, 9, 10, 5, 6, 0, 1, 1, 6, 13, 16, 15, 6, 7, 0, 1, 1, 7, 18, 26, 25, 21, 7, 8, 0, 1, 1, 8, 24, 40, 45, 36, 28, 8, 9, 0, 1, 1, 9, 31, 59, 75, 71, 49, 36, 9, 10, 0, 1, 1, 10, 39, 84, 120, 126, 105, 64, 45, 10, 11, 0, 1

Offset

1,8

Comments

T(n,n) = 1; T(n,n-1) = 0; T(n,n-2) = n-2;

T(n,n-3) = n-3; T(n,n-4) = (n-4)(n-3)/2; T(n,n-5) = (n-5)^2.

References

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

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

Links

Alois P. Heinz, Rows n = 1..141, flattened

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003 (see Table 3).

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Formula

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

For n<p+1, the number of compositions of n is binomial(n-1,k-1), except in the case of compositions of p into 1 part, which number equals 0. - Milan Janjic, Aug 06 2015

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

G.f.: [(x-x^2+x^3)/(1-x)]^k=sum{n>0, T(n,k)*x^n}, T(n,k)=T(n-1,k)+T(n-1,k-1)-T(n-2,k-1)+T(n-3,k-1). - Vladimir Kruchinin, Sep 29 2014

Example

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

Maple

p:= 2: 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 := 2: 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 := 2: 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 = 2; 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 *)

Prog

(Maxima)
T(n, k):=if n<0 then 0 else if n=k then 1 else if k=0 then 0 else  T(n-1, k)+T(n-1, k-1)-T(n-2, k-1)+T(n-3, k-1); /* Vladimir Kruchinin, Sep 23 2014 */

Crossrefs

Cf. A011973, A180178, A180179, A180180, A180181, A180182, A180183.

Cf. A097230 (same sequence with rows reversed).

Sequence in context: A343887 A215075 A287417 * A321196 A104578 A316827

Adjacent sequences: A180174 A180175 A180176 * A180178 A180179 A180180

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 15 2010

Status

approved