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 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. 11
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 n0, 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: A141169 A215075 A287417 * A104578 A316827 A286628 Adjacent sequences:  A180174 A180175 A180176 * A180178 A180179 A180180 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Aug 15 2010 STATUS approved

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Last modified October 23 04:22 EDT 2018. Contains 316519 sequences. (Running on oeis4.)