A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 7, 5, 1, 2, 4, 8, 11, 6, 1, 2, 4, 8, 15, 16, 7, 1, 2, 4, 8, 16, 27, 22, 8, 1, 2, 4, 8, 16, 31, 47, 29, 9, 1, 2, 4, 8, 16, 32, 59, 80, 37, 10, 1, 2, 4, 8, 16, 32, 63, 111, 134, 46, 11, 1, 2, 4, 8, 16, 32, 64, 123, 207, 222, 56, 12, 1, 2, 4, 8, 16, 32, 64, 127, 239, 384, 365, 67, 13

Offset

0,3

Links

Alois P. Heinz, Antidiagonals n = 0..140

Formula

G.f. of column k: (1-x+x^(k+1))/(1-3*x+2*x^2+x^(k+1)-x^(k+2)).

Example

A(4,1) = 11, because 11 binary words of length 4 have fewer than 1 0-digit between any pair of consecutive 1-digits: 0000, 0001, 0010, 0100, 1000, 0011, 0110, 1100, 0111, 1110, 1111.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2, ...
  3,  4,  4,  4,  4,  4, ...
  4,  7,  8,  8,  8,  8, ...
  5, 11, 15, 16, 16, 16, ...
  6, 16, 27, 31, 32, 32, ...

Maple

f:= proc(n, k) option remember; local j; if n=0 then 1 elif n<=k then 2^(n-1) else add(f(n-j, k), j=1..k) fi end: g:= proc(n, k) option remember; if n<0 then 0 else g(n-1, k) +f(n, k) fi end: A:= (n, k)-> `if`(n=0, g(0, k), A(n-1, k) +g(n-1, k)): seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

a[n_, k_] := SeriesCoefficient[(1 - x + x^(k+1))/(1 - 3*x + 2*x^2 + x^(k+1) - x^(k+2)), {x, 0, n}]; a[0, _] = 1; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)

Crossrefs

Columns 0-9 give: A000027(n+1), A000124, A000126(n+1), A007800(n+1), A145112, A145113, A145114, A145115, A145116, A145117.

Main diagonal gives A000079.

Cf. A141539.

Sequence in context: A080772 A259324 A216274 * A104795 A347570 A116925

Adjacent sequences: A145108 A145109 A145110 * A145112 A145113 A145114

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 02 2008

Status

approved