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 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. 6
 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 (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 7). 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. 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(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

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Last modified October 16 15:51 EDT 2019. Contains 328101 sequences. (Running on oeis4.)