A309400 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 5, 5, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11

Offset: 1

Author

Omar E. Pol, Nov 30 2019

Keywords

• nonn
• tabf

Comments

The number of parts in row n equals sigma(n) = A000203(n), the sum of the divisors of n. More generally, the number of parts congruent to 0 (mod m) in row m*n equals sigma(n).

The number of parts greater than 1 in row n equals A001065(n), the sum of the aliquot parts of n.

The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.

The number of partitions in row n equals A000005(n), the number of divisors of n.

The number of partitions in row n with an odd number of parts equals A001227(n).

The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).

The sum of row n equals n*A000005(n) = A038040(n).

Records in row n give the n-th row of A027750.

First n rows contain A000217(n) 1's.

The number of k's in row n is A126988(n,k).

The number of odd parts in row n is A002131(n).

The k-th block in row n has A056538(n,k) parts.

Column 1 gives A000012.

Right border gives A000027.

Examples

			Triangle begins:
[1];
[1,1], [2];
[1,1,1], [3];
[1,1,1,1], [2,2], [4];
[1,1,1,1,1], [5];
[1,1,1,1,1,1], [2,2,2], [3,3], [6];
[1,1,1,1,1,1,1], [7];
[1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8];
[1,1,1,1,1,1,1,1,1], [3,3,3], [9];
[1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10];
[1,1,1,1,1,1,1,1,1,1,1], [11];
[1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [12];
[1,1,1,1,1,1,1,1,1,1,1,1,1], [13];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2], [7,7], [14];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [3,3,3,3,3], [5,5,5], [15];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2,2], [4,4,4,4], [8,8], [16];
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Crossrefs

Mirror of A244051.

Cf. A000005, A000012, A000027, A000203, A000217, A001065, A001227, A002131, A027750, A038040, A048050, A056538, A126988, A237593, A245579, A299765, A328365.