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.

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, 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, 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

Offset: 1

Author

Omar E. Pol, Mar 18 2022

Keywords

• nonn
• tabf

Comments

Column k lists 2*k-1 interleaved with 2*k-2 zeros, and the first element of column k is in row A000384(k).

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

Conjecture 1: row sums give A341309.

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

Examples

			Triangle begins:
-----------------------
   n / k   1  2  3  4
-----------------------
   1 |     1;
   2 |     1;
   3 |     1;
   4 |     1;
   5 |     1;
   6 |     1, 3;
   7 |     1, 0;
   8 |     1, 0;
   9 |     1, 3;
  10 |     1, 0;
  11 |     1, 0;
  12 |     1, 3;
  13 |     1, 0;
  14 |     1, 0;
  15 |     1, 3, 5;
  16 |     1, 0, 0;
  17 |     1, 0, 0;
  18 |     1, 3, 0;
  19 |     1, 0, 0;
  20 |     1, 0, 5;
  21 |     1, 3, 0;
  22 |     1, 0, 0;
  23 |     1, 0, 0;
  24 |     1, 3, 0;
  25 |     1, 0, 5;
  26 |     1, 0, 0;
  27 |     1, 3, 0;
  28 |     1, 0, 0, 7;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10490 (rows 1..800 of triangle, flattened).

Crossrefs

Row lengths give A351846.

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

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

Programs

Mathematica

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 *)

Formula

T(n,k) = (2*k-1)*A351824(n,k). [Corrected by Paolo Xausa, Apr 09 2023]

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