A388717 Triangle read by rows: T(n,k) is the total number of runs of 1's in all length n binary words in which 1's occur in runs of at least k.

1, 3, 1, 8, 3, 1, 20, 6, 3, 1, 48, 12, 6, 3, 1, 112, 25, 10, 6, 3, 1, 256, 51, 17, 10, 6, 3, 1, 576, 101, 31, 15, 10, 6, 3, 1, 1280, 197, 58, 23, 15, 10, 6, 3, 1, 2816, 381, 106, 38, 21, 15, 10, 6, 3, 1, 6144, 731, 188, 66, 30, 21, 15, 10, 6, 3, 1, 13312, 1392, 328, 115, 46, 28, 21, 15, 10, 6, 3, 1

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Formula

T(n,k) = Sum_{i=1..floor((n+1)/(k+1))} binomial(n+1-(k-1)*i,2*i).

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

Example

Triangle begins:
     1;
     3,   1;
     8,   3,   1;
    20,   6,   3,  1;
    48,  12,   6,  3,  1;
   112,  25,  10,  6,  3,  1;
   256,  51,  17, 10,  6,  3,  1;
   576, 101,  31, 15, 10,  6,  3, 1;
  1280, 197,  58, 23, 15, 10,  6, 3, 1;
  2816, 381, 106, 38, 21, 15, 10, 6, 3, 1;
  ...

Prog

(PARI) T(n, k) = sum(i=1, (n+1)\(k+1), i*binomial(n+1-(k-1)*i, 2*i))

Crossrefs

Columns k=1..3 are A001792(n-1), A136444(n+1), A387573.

Cf. A000217, A388146, A388547.

Sequence in context: A249757 A207609 A322428 * A130300 A366873 A345656

Adjacent sequences: A388714 A388715 A388716 * A388718 A388719 A388720

Keyword

nonn,tabl

Author

Andrew Howroyd, Sep 19 2025

Status

approved