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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)].

%I A349042 #38 Nov 12 2021 12:25:55
%S A349042 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,
%T A349042 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,
%U A349042 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
%N 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)].
%C A349042 For n >= 1 the n-th row length equals A049820(n) + 1. The same definition, but for k from [0..n] gives A032531.
%H A349042 Michael De Vlieger, <a href="/A349042/a349042.png">Scatterplot of a(n)</a> for n=0..31686, i.e., rows 0..256.
%H A349042 Michael De Vlieger, <a href="/A349042/a349042_1.png">Scatterplot of a(n)</a>for n=0..518562, i.e., rows 0..1024
%H A349042 <a href="/index/In#inventory">Index entries for sequences related to the inventory sequence</a>
%e A349042 Triangle begins:
%e A349042       k=0  1  2  3  4  5
%e A349042   n=0:  0;
%e A349042   n=1:  1;
%e A349042   n=2:  1;
%e A349042   n=3:  1, 3;
%e A349042   n=4:  1, 4;
%e A349042   n=5:  1, 5, 0, 1;
%e A349042   n=6:  2, 6, 1;
%e A349042   n=7:  2, 7, 2, 1, 1, 1;
%t A349042 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 *)
%o A349042 (Python)
%o A349042 from collections import Counter
%o A349042 from sympy import divisor_count
%o A349042 def auptor(rows):
%o A349042     alst, inventory = [0], Counter([0])
%o A349042     for m in range(1, rows):
%o A349042         for k in range(m-divisor_count(m)+1):
%o A349042             c = inventory[k]; alst.append(c); inventory.update([c])
%o A349042     return alst
%o A349042 print(auptor(16)) # _Michael S. Branicky_, Nov 07 2021
%Y A349042 Cf. A032531, A049820.
%K A349042 nonn,tabf
%O A349042 0,5
%A A349042 _Ctibor O. Zizka_, Nov 06 2021