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A180178 Triangle read by rows: T(n,k) is the number of compositions of n without 3's and having k parts; 1 <= k <= n.

%I A180178 #15 Nov 15 2019 21:30:10
%S A180178 1,1,1,0,2,1,1,1,3,1,1,2,3,4,1,1,4,4,6,5,1,1,4,9,8,10,6,1,1,5,12,17,
%T A180178 15,15,7,1,1,6,15,28,30,26,21,8,1,1,7,21,38,56,51,42,28,9,1,1,8,27,56,
%U A180178 85,102,84,64,36,10,1,1,9,34,80,130,172,175,134,93,45,11,1,1,10,42,108,200
%N A180178 Triangle read by rows: T(n,k) is the number of compositions of n without 3's and having k parts; 1 <= k <= n.
%D A180178 P. Chinn and S. Heubach, Compositions of n with no occurrence of k, Congressus Numerantium, 164 (2003), pp. 33-51 (see Table 5).
%D A180178 R.P. Grimaldi, Compositions without the summand 1, Congressus Numerantium, 152, 2001, 33-43.
%H A180178 P. Chinn and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.
%F A180178 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).
%F A180178 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.
%e A180178 T(7,4)=8 because we have (1,2,2,2), (2,1,2,2), (2,2,1,2), (2,2,2,1), (1,1,1,4), (1,1,4,1), (1,4,1,1), and (4,1,1,1).
%e A180178 Triangle starts:
%e A180178   1;
%e A180178   1, 1;
%e A180178   0, 2, 1;
%e A180178   1, 1, 3, 1;
%e A180178   1, 2, 3, 4, 1;
%e A180178   1, 4, 4, 6, 5, 1;
%p A180178 p := 3: 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 A180178 p := 3: 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
%p A180178 with(combinat): m := 3: 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
%t A180178 p = 3; 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 *)
%Y A180178 Cf. A011973, A180177, A180179, A180180, A180181, A180182, A180183.
%K A180178 nonn,tabl
%O A180178 1,5
%A A180178 _Emeric Deutsch_, Aug 15 2010