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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.

%I A261019 #55 Oct 30 2024 19:41:21
%S A261019 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,
%T A261019 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,
%U A261019 40,12,1,1,9,142,212,274,32,185,116,52,1,1,10,231,354,509,64,408,296,168,2,4
%N 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.
%C A261019 This is a more compact version of the triangle in A261015, ending each row at the last nonzero entry.
%H A261019 Alois P. Heinz, N. J. A. Sloane, R. Zumkeller and Hiroaki Yamanouchi, <a href="/A261019/b261019.txt">Rows n = 1..58, flattened</a> (rows 17..25 from R. Zumkeller, rows 26..36 from Alois P. Heinz)
%H A261019 R. Zumkeller, <a href="/A261019/a261019.txt">The first 25 rows of the triangle, displayed as a triangle</a> (similar to the way the rows are shown in the Example section, but showing 25 rows).
%e A261019 The first 16 rows are:
%e A261019 1, 1,
%e A261019 1, 1,  1,    1,
%e A261019 1, 1,  2,    3,    1,
%e A261019 1, 1,  3,    6,    4,    1,
%e A261019 1, 1,  4,   11,   10,    5,
%e A261019 1, 1,  5,   19,   21,   15,    0,     2,
%e A261019 1, 1,  6,   32,   40,   35,    2,     9,     2,
%e A261019 1, 1,  7,   53,   72,   73,    6,    31,    10,     2,
%e A261019 1, 1,  8,   87,  125,  144,   15,    79,    40,    12,
%e A261019 1, 1,  9,  142,  212,  274,   32,   185,   116,    52,
%e A261019 1, 1, 10,  231,  354,  509,   64,   408,   296,   168,    2,    4,
%e A261019 1, 1, 11,  375,  585,  931,  120,   864,   699,   461,   24,   24,
%e A261019 1, 1, 12,  608,  960, 1685,  218,  1771,  1557,  1133,  130,  110,   2,   4,
%e A261019 1, 1, 13,  985, 1568, 3027,  385,  3555,  3325,  2612,  471,  387,  14,  24,  0,  16,
%e A261019 1, 1, 14, 1595, 2553, 5409,  668,  7021,  6893,  5759, 1401, 1135,  92, 120,  0,  90, 16,
%e A261019 1, 1, 15, 2582, 4148, 9628, 1142, 13696, 13964, 12309, 3734, 2972, 373, 439, 28, 390, 98, 16,
%e A261019 ...
%o A261019 (Haskell)
%o A261019 import Data.List (isInfixOf, sort, group)
%o A261019 a261019 n k = a261019_tabf !! (n-1) !! k
%o A261019 a261019_row n = a261019_tabf !! (n-1)
%o A261019 a261019_tabf = map (i 0 . group . sort . map f) a076478_tabf
%o A261019    where f bs = g a030308_tabf where
%o A261019            g (cs:css) | isInfixOf cs bs = g css
%o A261019                       | otherwise = foldr (\d v -> 2 * v + d) 0 cs
%o A261019          i _ [] = []
%o A261019          i r gss'@(gs:gss) | head gs == r = (length gs) : i (r + 1) gss
%o A261019                            | otherwise    = 0 : i (r + 1) gss'
%o A261019 -- _Reinhard Zumkeller_, Aug 18 2015
%Y A261019 Cf. A261015, A261016.
%Y A261019 The row lengths are given by A261017.
%Y A261019 Columns k=3-10 give: A001911, A001891, A261441, A261442, A261443, A261473, A261474, A261475.
%Y A261019 Cf. A076478, A030308, A000079 (row sums), A261392 (max per row).
%K A261019 nonn,tabf
%O A261019 1,9
%A A261019 _Alois P. Heinz_ and _N. J. A. Sloane_, Aug 17 2015