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

1,9

Comments

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

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).

Example

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,
...

Prog

(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

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).

Sequence in context: A275738 A202603 A279612 * A140737 A354009 A108756

Adjacent sequences: A261016 A261017 A261018 * A261020 A261021 A261022

Keyword

nonn,tabf

Author

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

Status

approved