login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A180182 Triangle read by rows: T(n,k) is the number of compositions of n without 7's and having k parts; 1<=k<=n. 7
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 1, 5, 21, 35, 35, 21, 7, 1, 1, 6, 25, 56, 70, 56, 28, 8, 1, 1, 7, 30, 80, 126, 126, 84, 36, 9, 1, 1, 8, 36, 108, 205, 252, 210, 120, 45, 10, 1, 1, 9, 43, 141, 310, 456, 462, 330, 165, 55, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
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 9).

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((-1)^{k-j}*binom(k,j)*binom(n-pk+pj-1,j-1), j=(pk-n)/(p-1)..k).

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.

EXAMPLE

T(10,2)=7 because we have (1,9), (9,1), (2,8), (8,2), (6,4), (4,6), and (5,5).

Triangle starts:

1;

1,1;

1,2,1;

1,3,3,1;

1,4,6,4,1;

1,5,10,10,5,1;

0,6,15,20,15,6,1;

MAPLE

p := 7: 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 := 7: 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 := 7: 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 = 7; 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-Fran├žois Alcover, Jan 28 2014, after Maple *)

CROSSREFS

Cf. A011973, A180177, A180178, A180179, A180180, A180181, A180183.

Sequence in context: A223968 A214846 A061676 * A275198 A095145 A095144

Adjacent sequences:  A180179 A180180 A180181 * A180183 A180184 A180185

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 15 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 12 15:01 EST 2017. Contains 295939 sequences.