A351824 Irregular triangle read by rows: T(n,k) is the number of partitions of n into 2*k-1 consecutive parts, n >= 1, k >= 1. Column k lists 1's interleaved with 2*k-2 zeros, and the first element of column k is in row A000384(k).

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

Offset

1

Comments

Conjecture 1: T(n,k) is the number of subparts, in an octant of the symmetric representation of sigma(n), that arises from the (2*k-1)-th double-staircase of the double-staircases diagram of n described in A335616.

Conjecture 2: Indices of 1's coincide with indices of nonzero terms in A347263, A347529, A351819.

For the above conjectures see also the "ziggurat" diagram described in A347186.

This triangle is formed by the odd-indexed columns of the triangle A237048.

Terms can be 0 or 1.

Links

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

Formula

T(n,k) = A352499(n,k)/n. - Omar E. Pol, Mar 24 2022

T(n,k) = [(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - Paolo Xausa, Apr 01 2023

Example

Triangle begins:
-----------------------
   n / k   1  2  3  4
-----------------------
   1 |     1;
   2 |     1;
   3 |     1;
   4 |     1;
   5 |     1;
   6 |     1, 1;
   7 |     1, 0;
   8 |     1, 0;
   9 |     1, 1;
  10 |     1, 0;
  11 |     1, 0;
  12 |     1, 1;
  13 |     1, 0;
  14 |     1, 0;
  15 |     1, 1, 1;
  16 |     1, 0, 0;
  17 |     1, 0, 0;
  18 |     1, 1, 0;
  19 |     1, 0, 0;
  20 |     1, 0, 1;
  21 |     1, 1, 0;
  22 |     1, 0, 0;
  23 |     1, 0, 0;
  24 |     1, 1, 0;
  25 |     1, 0, 1;
  26 |     1, 0, 0;
  27 |     1, 1, 0;
  28 |     1, 0, 0, 1;
  ...
For n = 15 the partitions of 15 into an odd number of consecutive parts are [15], [6, 5, 4] and [5, 4, 3, 2, 1]. There are a partition with only one part, a partition with three parts and a partition with five parts, so the 15th row of triangle is [1, 1, 1].

Mathematica

A351824[rowmax_]:=Table[Boole[Divisible[n, 2k-1]], {n, rowmax}, {k, Floor[(Sqrt[8n+1]+1)/4]}]; A351824[50] (* Paolo Xausa, Apr 01 2023 *)

Crossrefs

Row sums give A082647.

Row n has length A351846(n).

Cf. A000384, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244250, A262619, A262626, A279387, A280850, A280851, A286000, A286001, A296508, A299765, A335616, A347186, A347263, A347529, A348854, A351819, A352257, A352499.

Sequence in context: A248863 A328306 A267256 * A365605 A365716 A334460

Adjacent sequences: A351821 A351822 A351823 * A351825 A351826 A351827

Keyword

nonn,tabf,easy

Author

Omar E. Pol, Feb 20 2022

Status

approved