A349042 Triangle read by rows in which row n >= 1 lists the count of 0's, ..., k's in all previous terms in the triangle. T(0,0) = 0, k is from [0..A049820(n)].

0, 1, 1, 1, 3, 1, 4, 1, 5, 0, 1, 2, 6, 1, 2, 7, 2, 1, 1, 1, 2, 10, 4, 1, 2, 2, 11, 6, 1, 2, 1, 2, 2, 13, 9, 1, 2, 1, 2, 2, 15, 12, 1, 2, 1, 2, 1, 0, 1, 3, 19, 14, 2, 2, 1, 2, 3, 20, 17, 3, 2, 1, 2, 1, 0, 1, 1, 1, 4, 25, 19, 4, 4, 1, 2, 1, 0, 1, 1, 5, 29, 20, 4, 6, 2, 3, 1, 0, 1, 1, 1

Offset: 0

Author

Ctibor O. Zizka, Nov 06 2021

Keywords

• nonn
• tabf

Comments

For n >= 1 the n-th row length equals A049820(n) + 1. The same definition, but for k from [0..n] gives A032531.

Examples

			Triangle begins:
      k=0  1  2  3  4  5
  n=0:  0;
  n=1:  1;
  n=2:  1;
  n=3:  1, 3;
  n=4:  1, 4;
  n=5:  1, 5, 0, 1;
  n=6:  2, 6, 1;
  n=7:  2, 7, 2, 1, 1, 1;
		

Links

• Michael De Vlieger, Scatterplot of a(n) for n=0..31686, i.e., rows 0..256.
• Michael De Vlieger, Scatterplot of a(n)for n=0..518562, i.e., rows 0..1024
• Index entries for sequences related to the inventory sequence

Crossrefs

Cf. A032531, A049820.

Programs

Mathematica

c[] = 0; Reap[Do[w = {}; Array[(Set[m, c[#]]; c[m]++; AppendTo[w, m]) &, If[n == 0, 1, n - DivisorSigma[0, n] + 1], 0]; Sow[w], {n, 0, 15}]][[-1, -1]] // Flatten (* _Michael De Vlieger, Nov 09 2021 *)

Python

from collections import Counter
from sympy import divisor_count
def auptor(rows):
    alst, inventory = [0], Counter([0])
    for m in range(1, rows):
        for k in range(m-divisor_count(m)+1):
            c = inventory[k]; alst.append(c); inventory.update([c])
    return alst
print(auptor(16)) # Michael S. Branicky, Nov 07 2021