A354009 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of equal parts, in nonincreasing order.

1, 2, 3, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 2, 2, 2, 7, 1, 1, 1, 1, 1, 1, 1, 8, 9, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 2, 2, 2, 2, 2, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 4, 4, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 2, 2, 2, 2, 2, 2, 2, 15, 5, 5, 5, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

The number of partitions in row n equals A001227(n), the number of odd divisors of n, and equals the number of partitions of n into consecutive parts, and equals the number of subparts in the symmetric representation of sigma(n).

The sum of row n equals A245579(n), the sum of all parts of all partitions of n into consecutive parts.

The length of row n equals A000593(n), the sum of the odd divisors of n.

Row n has length 1 if and only if n is a power of 2.

Is the right border the same as A006519?

Links

Table of n, a(n) for n=1..107.

Example

Triangle begins:
   [1];
   [2];
   [3], [1,1,1];
   [4];
   [5], [1,1,1,1,1];
   [6], [2,2,2];
   [7], [1,1,1,1,1,1,1];
   [8];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [2,2,2,2,2];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [4,4,4];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [2,2,2,2,2,2,2];
  [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [16];
  ...
For n = 10, in row 10 there are two partitions, equaling the number of odd divisors of 10, they are [1, 5], and equaling the number of partitions of 10 into consecutive parts, they are [10], [4, 3, 2, 1], and equaling the number of subparts in the symmetric representation of sigma(10), they are [9, 9].
The sum of row 10 is [10] + [2 + 2 + 2 + 2 + 2] = 20 equaling the sum of all parts of all partitions of 10 into consecutive parts, that is [10] + [4 + 3 + 2 + 1] = 20.
The length of row 10 is equal to 6 equaling the sum of the odd divisors of 10, that is 1 + 5 = 6.

Mathematica

Table[ConstantArray[n/#, #] & /@ Select[Divisors[n], OddQ], {n, 15}] // Flatten (* Michael De Vlieger, Jul 15 2022 *)

Prog

(PARI) row(n) = my(v=[]); fordiv(n, d, if ((n/d)%2, v = concat(v, vector(n/d, k, d)))); Vecrev(v); \\ Michel Marcus, Jul 16 2022

Crossrefs

Subsequence of A244051.

The number of partitions in row n equals A001227(n).

Row lengths give A000593.

Row sums give A245579.

Column 1 gives A000027.

Cf. A000079, A000203, A006519, A237593, A279387 (subparts), A299765.

Sequence in context: A279612 A261019 A140737 * A108756 A106178 A305807

Adjacent sequences: A354006 A354007 A354008 * A354010 A354011 A354012

Keyword

nonn,tabf

Author

Omar E. Pol, Jul 13 2022

Status

approved