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

0,5

Links

Alois P. Heinz, Rows n = 0..140, flattened

Index entries for triangles generated by the Multiset Transformation

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A117372 A127570 A340583 * A212186 A274662 A186827

Adjacent sequences: A292503 A292504 A292505 * A292507 A292508 A292509

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 17 2017

Status

approved