A126027 -id:A126027 - cp's OEIS frontend

A333867 Table with T(1,1) = 1; for n>1, T(n,k) is the number of k's in rows 1 through n-1.

1, 1, 2, 2, 1, 3, 2, 3, 3, 1, 4, 3, 3, 4, 3, 5, 1, 5, 3, 6, 2, 1, 6, 4, 7, 2, 2, 1, 7, 6, 7, 3, 2, 2, 1, 8, 8, 8, 3, 2, 3, 3, 8, 9, 11, 3, 2, 3, 3, 3, 8, 10, 15, 3, 2, 3, 3, 4, 1, 0, 1, 10, 11, 18, 4, 2, 3, 3, 5, 1, 1, 1, 0, 0, 0, 1, 14, 12, 20, 5, 3, 3, 3, 5, 1, 2, 2, 0, 0, 0, 1, 0, 0, 1, 17, 14, 23, 5, 5

Offset: 1

Clark Kimberling

Equivalently, list 1, where, at stage k>1, write i in list 1 and j in list 2, where i is the number of j's in list 1, for j=1,2,...,m, where m=max number in list 1 from stages 1 to k-1; stage 1 is 1 in list 1.
Differs from A030717 in that this sequence includes 0's. - Sean A. Irvine, Apr 08 2020
Nevertheless, this sequence starts each row with the count of 1's, not 0's, whose counts are not recorded in the sequence (cf. A174382, which is also initialized with a 0). - Peter Munn, Oct 11 2022

			1;
1;
2;
2, 1;
3, 2;
3, 3, 1;
4, 3, 3;

Franklin T. Adams-Watters, First 32 rows, flattened
Michael De Vlieger, Plot a(w(j) + k - 1) at (j,k) for j = 1..60 and w the sequence of partial sums of A126027, showing a(m) = 0 in black, a(m) = 1 in red, and a(m) > 1 in light blue.
Sean A. Irvine, Java program (github)

Cf. A126027 (row lengths), A006920, A030717 (zeros suppressed).
Cf. A174382.
Cf. A253170 (row sums).

import Data.List (sort, group)
a030717 n k = a030717_tabf !! (n-1) !! (k-1)
a030717_row n = a030717_tabf !! (n-1)
a030717_tabf = [1] : f [1] where
   f xs = ys : f ((filter (> 0) ys) ++ xs) where
          ys = h (group $ sort xs) [1..] where
               h [] _ = []
               h vss'@(vs:vss) (w:ws)
                 | head vs == w = (length vs) : h vss ws
                 | otherwise    = 0 : h vss' ws
-- Reinhard Zumkeller, Dec 28 2014

t = {{1}}; Do[AppendTo[t, BinCounts[#, {1, Max[#] + 1}] &[Flatten[t]]], {30}];
Map[Length, t] (* A126027*)
Map[Total, t]  (* A253170*)
Flatten[t]     (* A333867 *)  (* Peter J. C. Moses, Apr 09 2020 *)

More terms from Franklin T. Adams-Watters, Dec 14 2006

A253170 Row sums of table A333867.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 42, 50, 60, 72, 85, 101, 119, 139, 162, 187, 215, 246, 279, 315, 353, 394, 437, 482, 529, 579, 632, 688, 748, 810, 875, 944, 1016, 1094, 1177, 1263, 1351, 1443, 1538, 1637, 1742, 1852, 1966, 2083, 2205, 2330, 2460

Offset: 1

Reinhard Zumkeller, Dec 28 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Cf. A333867, A126027, A030719 (first differences).

Haskell
```
a253170 = sum . a333867_row
```

A240508 Row lengths of table A174382.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 13, 17, 20, 23, 29, 38, 52, 73, 106, 157, 239, 369, 579, 916, 1460, 2338, 3757, 6051, 9761, 15762, 25470, 41176, 66587, 107701, 174223, 281856, 456008, 737790, 1193721, 1931431, 3125070, 5056416

Offset: 1

Reinhard Zumkeller, Apr 06 2014

nonn

Cf. A126027.

Haskell
```
a240508 = length . a174382_row
```

cp's OEIS Frontend

A333867 Table with T(1,1) = 1; for n>1, T(n,k) is the number of k's in rows 1 through n-1.

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A253170 Row sums of table A333867.

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A240508 Row lengths of table A174382.

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