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 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. 8
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 A116925 A210950 Adjacent sequences:  A145108 A145109 A145110 * A145112 A145113 A145114 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Oct 02 2008 STATUS approved

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