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 A352269 #54 Apr 12 2023 08:06:20
%S A352269 1,1,1,1,1,1,3,1,0,1,0,1,3,1,0,1,0,1,3,1,0,1,0,1,3,5,1,0,0,1,0,0,1,3,
%T A352269 0,1,0,0,1,0,5,1,3,0,1,0,0,1,0,0,1,3,0,1,0,5,1,0,0,1,3,0,1,0,0,7,1,0,
%U A352269 0,0,1,3,5,0,1,0,0,0,1,0,0,0,1,3,0,0,1,0,0,0,1,0,5,7,1,3,0,0,1,0,0,0,1,0,0,0
%N A352269 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into 2*k-1 consecutive parts, 1 <= k <= A351846(n). T(n,k) = 0 if no such partition exists.
%C A352269 Column k lists 2*k-1 interleaved with 2*k-2 zeros, and the first element of column k is in row A000384(k).
%C A352269 This triangle is formed from the odd-indexed columns of the triangle A285914.
%C A352269 Conjecture 1: row sums give A341309.
%C A352269 Conjecture 2: A347529, A351819, A347263 have the same indices of zero and nonzero terms as A351824 and this sequence.
%H A352269 Paolo Xausa, <a href="/A352269/b352269.txt">Table of n, a(n) for n = 1..10490</a> (rows 1..800 of triangle, flattened).
%F A352269 T(n,k) = (2*k-1)*A351824(n,k). [Corrected by _Paolo Xausa_, Apr 09 2023]
%F A352269 T(n,k) = (2*k-1)*[(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - _Paolo Xausa_, Apr 09 2023
%e A352269 Triangle begins:
%e A352269 -----------------------
%e A352269 n / k 1 2 3 4
%e A352269 -----------------------
%e A352269 1 | 1;
%e A352269 2 | 1;
%e A352269 3 | 1;
%e A352269 4 | 1;
%e A352269 5 | 1;
%e A352269 6 | 1, 3;
%e A352269 7 | 1, 0;
%e A352269 8 | 1, 0;
%e A352269 9 | 1, 3;
%e A352269 10 | 1, 0;
%e A352269 11 | 1, 0;
%e A352269 12 | 1, 3;
%e A352269 13 | 1, 0;
%e A352269 14 | 1, 0;
%e A352269 15 | 1, 3, 5;
%e A352269 16 | 1, 0, 0;
%e A352269 17 | 1, 0, 0;
%e A352269 18 | 1, 3, 0;
%e A352269 19 | 1, 0, 0;
%e A352269 20 | 1, 0, 5;
%e A352269 21 | 1, 3, 0;
%e A352269 22 | 1, 0, 0;
%e A352269 23 | 1, 0, 0;
%e A352269 24 | 1, 3, 0;
%e A352269 25 | 1, 0, 5;
%e A352269 26 | 1, 0, 0;
%e A352269 27 | 1, 3, 0;
%e A352269 28 | 1, 0, 0, 7;
%e A352269 ...
%t A352269 A352269[rowmax_]:=Table[If[Divisible[n,2k-1],2k-1,0],{n,rowmax},{k,Floor[(Sqrt[8n+1]+1)/4]}];A352269[50] (* _Paolo Xausa_, Apr 09 2023 *)
%Y A352269 Row lengths give A351846.
%Y A352269 The number of nonzeros terms in row n equals A082647(n).
%Y A352269 Cf. A000384, A285914, A299765, A341309, A347529, A351819, A347263, A351824, A352425.
%K A352269 nonn,tabf
%O A352269 1,7
%A A352269 _Omar E. Pol_, Mar 18 2022