This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A176206 #107 Jun 04 2022 14:03:00
%S A176206 1,2,1,3,2,1,1,4,3,2,2,1,1,1,5,4,3,3,2,2,2,1,1,1,1,1,6,5,4,4,3,3,3,2,
%T A176206 2,2,2,2,1,1,1,1,1,1,1,7,6,5,5,4,4,4,3,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,
%U A176206 1,1,1,1,1,1,1,8,7,6,6,5,5,5,4,4,4,4,4,3,3,3,3,3,3,3
%N A176206 Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: row n has length A000070(n-1) and every column k gives the positive integers.
%C A176206 The original definition was: An irregular table: Row n begins with n, counts down to 1 and repeats the intermediate numbers as often as given by the partition numbers.
%C A176206 Row n contains a decreasing sequence where n-k is repeated A000041(k) times, k = 0..n-1.
%C A176206 From _Omar E. Pol_, Nov 23 2020: (Start)
%C A176206 Row n lists in nonincreasing order the first A000070(n-1) terms of A336811.
%C A176206 In other words: row n lists in nonincreasing order the terms from the first n rows of triangle A336811.
%C A176206 Conjecture: all divisors of all terms in row n are also all parts of all partitions of n.
%C A176206 For more information see the example and A336811 which contains the most elementary conjecture about the correspondence divisors/partitions.
%C A176206 Row sums give A014153.
%C A176206 A338156 lists the divisors of every term of this sequence.
%C A176206 The n-th row of A340581 lists in nonincreasing order the terms of the first n rows of this triangle.
%C A176206 For a regular triangle with the same row sums see A141157. (End)
%C A176206 From _Omar E. Pol_, Jul 31 2021: (Start)
%C A176206 The number of k's in row n is equal to A000041(n-k), 1 <= k <= n.
%C A176206 The number of terms >= k in row n is equal to A000070(n-k), 1 <= k <= n.
%C A176206 The number of k's in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.
%C A176206 The number of terms >= k in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A014153(n-k), 1 <= k <= n. (End)
%H A176206 Paolo Xausa, <a href="/A176206/b176206.txt">Table of n, a(n) for n = 1..10980 (rows 1..21 of triangle, flattened)</a>
%e A176206 Triangle begins:
%e A176206 1;
%e A176206 2, 1;
%e A176206 3, 2, 1, 1;
%e A176206 4, 3, 2, 2, 1, 1, 1;
%e A176206 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1;
%e A176206 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
%e A176206 7, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, ...
%e A176206 ... Extended by _Omar E. Pol_, Nov 23 2020
%e A176206 From _Omar E. Pol_, Jan 25 2020: (Start)
%e A176206 For n = 5, by definition the length of row 5 is A000070(5-1) = A000070(4) = 12, so the row 5 of triangle has 12 terms. Since every column lists the positive integers A000027 so the row 5 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1].
%e A176206 Then we have that the divisors of the numbers of the 5th row are:
%e A176206 .
%e A176206 5th row of triangle -----> 5 4 3 3 2 2 2 1 1 1 1 1
%e A176206 1 2 1 1 1 1 1
%e A176206 1
%e A176206 .
%e A176206 There are twelve 1's, four 2's, two 3's, one 4 and one 5.
%e A176206 In total there are 12 + 4 + 2 + 1 + 1 = 20 divisors.
%e A176206 On the other hand the partitions of 5 are as shown below:
%e A176206 .
%e A176206 . 5
%e A176206 . 3 2
%e A176206 . 4 1
%e A176206 . 2 2 1
%e A176206 . 3 1 1
%e A176206 . 2 1 1 1
%e A176206 . 1 1 1 1 1
%e A176206 .
%e A176206 There are twelve 1's, four 2's, two 3's, one 4 and one 5, as shown also in the 5th row of triangle A066633.
%e A176206 In total there are 12 + 4 + 2 + 1 + 1 = A006128(5) = 20 parts.
%e A176206 Finally in accordance with the conjecture we can see that all divisors of all numbers in the 5th row of the triangle are the same positive integers as all parts of all partitions of 5. (End)
%t A176206 Table[Flatten[Table[ConstantArray[n-k,PartitionsP[k]],{k,0,n-1}]],{n,10}] (* _Paolo Xausa_, May 30 2022 *)
%Y A176206 Cf. A000027 (columns), A000070 (row lengths), A338156 (divisors), A340061 (mirror).
%Y A176206 Cf. A000041, A006128, A014153, A027750, A066633, A141157, A176207, A336811, A338156, A340581.
%K A176206 easy,nonn,tabf
%O A176206 1,2
%A A176206 _Alford Arnold_, Apr 11 2010
%E A176206 New name, changed offset, edited and more terms from _Omar E. Pol_, Nov 22 2020