A174382 T(1,0)=0 and for n > 1, T(n,k) is the number of k's in rows 1 to n - 1.

0, 1, 1, 1, 1, 3, 1, 4, 0, 1, 2, 6, 0, 1, 1, 3, 8, 1, 1, 1, 0, 1, 4, 12, 1, 2, 1, 0, 1, 0, 1, 6, 16, 2, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 11, 19, 5, 2, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 19, 22, 8, 2, 2, 1, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 27, 28, 11, 2, 2, 1, 2, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1

Offset: 1

Author

Paolo P. Lava & Giorgio Balzarotti, Mar 17 2010

Keywords

• easy
• nonn
• tabf
• look
• changed

Comments

Construction as in A333867 but starting with a 0 and including a count of 0s at the start of each row. [Edited by Peter Munn, Oct 11 2022]

See A342585 for a similarly defined sequence that has been analyzed more and has lists of other related sequences. - Peter Munn, Oct 08 2022

Examples

			0;
1; # one zero
1,1; # one zero, one one
1,3; # one zero, three ones
1,4,0,1; # one zero, four ones, zero twos, one three
		

Links

• Reinhard Zumkeller, Rows n = 1..25 of table, flattened

Crossrefs

Cf. A240508 (row lengths).

Cf. A333867, A342585.

Programs

Haskell

import Data.List (sort, group)
a174382 n k = a174382_tabf !! (n-1) !! k
a174382_row n = a174382_tabf !! (n-1)
a174382_tabf = iterate f [0] where
   f xs = g (xs ++ [0, 0 ..]) [0..] (map head zs) (map length zs)
     where g   _ [] = []
           g (u:us) (k:ks) hs'@(h:hs) vs'@(v:vs)
             | k == h = u + v : g us ks hs vs
             | k /= h = u : g us ks hs' vs'
           zs = group $ sort xs
-- Reinhard Zumkeller, Apr 06 2014

Maple

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+add(x^i, i=T(n))) end:
T:= proc(n) option remember; `if`(n=1, 0, (p->
      seq(coeff(p, x, i), i=0..degree(p)))(b(n-1)))
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 25 2025