|
%I #15 Nov 15 2019 21:30:41
%S 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,
%T 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,
%U 36,108,205,252,210,120,45,10,1,1,9,43,141,310,456,462,330,165,55,11,1
%N Triangle read by rows: T(n,k) is the number of compositions of n without 7's and having k parts; 1 <= k <= n.
%D P. Chinn and S. Heubach, Compositions of n with no occurrence of k, Congressus Numerantium, 164 (2003), pp. 33-51 (see Table 9).
%D R.P. Grimaldi, Compositions without the summand 1, Congressus Numerantium, 152, 2001, 33-43.
%H 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 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 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 T(10,2)=7 because we have (1,9), (9,1), (2,8), (8,2), (6,4), (4,6), and (5,5).
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 3, 1;
%e 1, 4, 6, 4, 1;
%e 1, 5, 10, 10, 5, 1;
%e 0, 6, 15, 20, 15, 6, 1;
%p 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 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
%p 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
%t 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-François Alcover_, Jan 28 2014, after Maple *)
%Y Cf. A011973, A180177, A180178, A180179, A180180, A180181, A180183.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Aug 15 2010
|
