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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140
Crossrefs
Programs
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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);
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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 *)
Formula
G.f. of column k: (1-x+x^(k+1))/(1-3*x+2*x^2+x^(k+1)-x^(k+2)).