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

Author

Omar E. Pol, Jul 13 2022

Keywords

• nonn
• tabf

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?

Examples

			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.
		

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.

Programs

Mathematica

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

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