A292506 Number T(n,k) of multisets of exactly k nonempty binary words with a total of n letters such that no word has a majority of 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 3, 1, 0, 4, 3, 1, 0, 11, 10, 3, 1, 0, 16, 23, 10, 3, 1, 0, 42, 59, 33, 10, 3, 1, 0, 64, 134, 83, 33, 10, 3, 1, 0, 163, 320, 230, 98, 33, 10, 3, 1, 0, 256, 699, 568, 270, 98, 33, 10, 3, 1, 0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1, 0, 1024, 3434, 3439, 1935, 798, 291, 98, 33, 10, 3, 1

Offset: 0

Author

Alois P. Heinz, Sep 17 2017

Keywords

• nonn
• tabl

Examples

			T(4,2) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,    1;
  0,   4,    3,    1;
  0,  11,   10,    3,   1;
  0,  16,   23,   10,   3,   1;
  0,  42,   59,   33,  10,   3,  1;
  0,  64,  134,   83,  33,  10,  3,  1;
  0, 163,  320,  230,  98,  33, 10,  3,  1;
  0, 256,  699,  568, 270,  98, 33, 10,  3, 1;
  0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Index entries for triangles generated by the Multiset Transformation

Crossrefs

Columns k=0-10 give: A000007, A027306 (for n>0), A316403, A316404, A316405, A316406, A316407, A316408, A316409, A316410, A316411.

Row sums give A292548.

T(2n,n) gives A292549.

Cf. A209406, A226873, A290222.

Programs

Maple

g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

Mathematica

g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)

Formula

G.f.: Product_{j>=1} 1/(1-y*x^j)^A027306(j).