A261019 Irregular triangle read by rows: T(n,k) (0 <= k <= A261017(n)) = number of binary strings of length n such that the smallest number whose binary representation is not visible in the string is k.

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 6, 4, 1, 1, 1, 4, 11, 10, 5, 1, 1, 5, 19, 21, 15, 0, 2, 1, 1, 6, 32, 40, 35, 2, 9, 2, 1, 1, 7, 53, 72, 73, 6, 31, 10, 2, 1, 1, 8, 87, 125, 144, 15, 79, 40, 12, 1, 1, 9, 142, 212, 274, 32, 185, 116, 52, 1, 1, 10, 231, 354, 509, 64, 408, 296, 168, 2, 4

Offset: 1

Author

Alois P. Heinz and N. J. A. Sloane, Aug 17 2015

Keywords

• nonn
• tabf

Comments

This is a more compact version of the triangle in A261015, ending each row at the last nonzero entry.

Examples

			The first 16 rows are:
1, 1,
1, 1,  1,    1,
1, 1,  2,    3,    1,
1, 1,  3,    6,    4,    1,
1, 1,  4,   11,   10,    5,
1, 1,  5,   19,   21,   15,    0,     2,
1, 1,  6,   32,   40,   35,    2,     9,     2,
1, 1,  7,   53,   72,   73,    6,    31,    10,     2,
1, 1,  8,   87,  125,  144,   15,    79,    40,    12,
1, 1,  9,  142,  212,  274,   32,   185,   116,    52,
1, 1, 10,  231,  354,  509,   64,   408,   296,   168,    2,    4,
1, 1, 11,  375,  585,  931,  120,   864,   699,   461,   24,   24,
1, 1, 12,  608,  960, 1685,  218,  1771,  1557,  1133,  130,  110,   2,   4,
1, 1, 13,  985, 1568, 3027,  385,  3555,  3325,  2612,  471,  387,  14,  24,  0,  16,
1, 1, 14, 1595, 2553, 5409,  668,  7021,  6893,  5759, 1401, 1135,  92, 120,  0,  90, 16,
1, 1, 15, 2582, 4148, 9628, 1142, 13696, 13964, 12309, 3734, 2972, 373, 439, 28, 390, 98, 16,
...
		

Links

• Alois P. Heinz, N. J. A. Sloane, R. Zumkeller and Hiroaki Yamanouchi, Rows n = 1..58, flattened (rows 17..25 from R. Zumkeller, rows 26..36 from Alois P. Heinz)
• R. Zumkeller, The first 25 rows of the triangle, displayed as a triangle (similar to the way the rows are shown in the Example section, but showing 25 rows).

Crossrefs

Cf. A261015, A261016.

The row lengths are given by A261017.

Columns k=3-10 give: A001911, A001891, A261441, A261442, A261443, A261473, A261474, A261475.

Cf. A076478, A030308, A000079 (row sums), A261392 (max per row).

Programs

Haskell

import Data.List (isInfixOf, sort, group)
a261019 n k = a261019_tabf !! (n-1) !! k
a261019_row n = a261019_tabf !! (n-1)
a261019_tabf = map (i 0 . group . sort . map f) a076478_tabf
   where f bs = g a030308_tabf where
           g (cs:css) | isInfixOf cs bs = g css
                      | otherwise = foldr (\d v -> 2 * v + d) 0 cs
         i _ [] = []
         i r gss'@(gs:gss) | head gs == r = (length gs) : i (r + 1) gss
                           | otherwise    = 0 : i (r + 1) gss'
-- Reinhard Zumkeller, Aug 18 2015