A345116 -id:A345116 - cp's OEIS frontend

A110730 Irregular triangle read by rows in which row n lists n 1's followed by (n-1) 2's followed by (n-3) 3's ... followed by 1 n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3

Offset: 1

Amarnath Murthy, Aug 09 2005

From Omar E. Pol, May 08 2021: (Start)
Row n has length A000217(n).
Row sums give A000292, n >= 1.
Row n lists the terms of the n-th row A333516 in nondecreasing order.
(End)

			From _Omar E. Pol_, May 08 2021: (Start)
Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 2, 3;
  1, 1, 1, 1, 2, 2, 2, 3, 3, 4;
  1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5;
  1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6;
  ...
(End)

Cf. A004736.

Cf. A000217, A000292, A333516, A345116 (mirror).

Flatten@Array[Table[Table[k,#-k+1],{k,#}]&,10] (* Giorgos Kalogeropoulos, Jun 08 2021 *)

Name clarified by Omar E. Pol, Jun 08 2021

A345272 Irregular triangle read by rows T(n,k) in which row n lists in nonincreasing order all divisors of the terms of the n-th row of triangle A110730, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jun 12 2021

Note that in the definition A110730 can be replaced with A333516 or with A345116 since these three triangles contain in every row the same terms but in distinct order.

The sum of n-th row is equal to A175254(n) equaling the volume (also the number of cubes) of the stepped pyramid with n levels described in A245092.

			Triangle begins:
1;
2, 1, 1, 1;
3, 2, 2, 1, 1, 1, 1, 1, 1;
4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...
For n = 3 the third row of A110730 is [1, 1, 1, 2, 2, 3], so the divisors of these terms in nonincreasing order are [3, 2, 2, 1, 1, 1, 1, 1, 1], the same as the third row of triangle.

Cf. A110730, A175254, A237593, A245092, A333516, A345116.

row(n) = my(v=[]); for (k=1, n, for (j=1, n-k+1, v = concat(v, divisors(k)))); vecsort(v,,4); \\ Michel Marcus, Jun 14 2021

cp's OEIS Frontend

A110730 Irregular triangle read by rows in which row n lists n 1's followed by (n-1) 2's followed by (n-3) 3's ... followed by 1 n.

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