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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.
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%I #54 Apr 12 2023 08:06:20

%S 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 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 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 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 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 This triangle is formed from the odd-indexed columns of the triangle A285914.

%C Conjecture 1: row sums give A341309.

%C Conjecture 2: A347529, A351819, A347263 have the same indices of zero and nonzero terms as A351824 and this sequence.

%H Paolo Xausa, <a href="/A352269/b352269.txt">Table of n, a(n) for n = 1..10490</a> (rows 1..800 of triangle, flattened).

%F T(n,k) = (2*k-1)*A351824(n,k). [Corrected by _Paolo Xausa_, Apr 09 2023]

%F 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 Triangle begins:

%e -----------------------

%e n / k 1 2 3 4

%e -----------------------

%e 1 | 1;

%e 2 | 1;

%e 3 | 1;

%e 4 | 1;

%e 5 | 1;

%e 6 | 1, 3;

%e 7 | 1, 0;

%e 8 | 1, 0;

%e 9 | 1, 3;

%e 10 | 1, 0;

%e 11 | 1, 0;

%e 12 | 1, 3;

%e 13 | 1, 0;

%e 14 | 1, 0;

%e 15 | 1, 3, 5;

%e 16 | 1, 0, 0;

%e 17 | 1, 0, 0;

%e 18 | 1, 3, 0;

%e 19 | 1, 0, 0;

%e 20 | 1, 0, 5;

%e 21 | 1, 3, 0;

%e 22 | 1, 0, 0;

%e 23 | 1, 0, 0;

%e 24 | 1, 3, 0;

%e 25 | 1, 0, 5;

%e 26 | 1, 0, 0;

%e 27 | 1, 3, 0;

%e 28 | 1, 0, 0, 7;

%e ...

%t 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 Row lengths give A351846.

%Y The number of nonzeros terms in row n equals A082647(n).

%Y Cf. A000384, A285914, A299765, A341309, A347529, A351819, A347263, A351824, A352425.

%K nonn,tabf

%O 1,7

%A _Omar E. Pol_, Mar 18 2022