A359279 Irregular triangle T(n,k) (n>=1, k>=1) read by rows in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive triangular numbers A000217.

1, 3, 6, 1, 10, 3, 1, 15, 6, 3, 1, 1, 21, 10, 6, 3, 3, 1, 1, 28, 15, 10, 6, 6, 3, 3, 1, 1, 1, 1, 36, 21, 15, 10, 10, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 45, 28, 21, 15, 15, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 55, 36, 28, 21, 21, 15, 15, 10, 10, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

All divisors of the largest partition into consecutive parts of all terms in row n are also all parts of all partitions of n.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened).

Formula

T(n,k) = A000217(A336811(n,k)).

Example

Triangle begins:
   1;
   3;
   6,  1;
  10,  3,  1;
  15,  6,  3,  1,  1;
  21, 10,  6,  3,  3,  1,  1;
  28, 15, 10,  6,  6,  3,  3, 1, 1, 1, 1;
  36, 21, 15, 10, 10,  6,  6, 3, 3, 3, 3, 1, 1, 1, 1;
  45, 28, 21, 15, 15, 10, 10, 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From Omar E. Pol, Feb 28 2023: (Start)
For n = 4 the fourth row is [10, 3, 1]. The largest partition into consecutive parts of every term are respectively [4, 3, 2, 1], [2, 1], [1]. The divisors of these parts are [(1, 2, 4), (1, 3), (1, 2), (1)], [(1, 2), (1)], [1]. These 12 divisors are also all parts of all partitions of 4. They are  [(4), (2, 2), (3, 1), (2, 1, 1), (1, 1, 1, 1)]. (End)

Mathematica

A359279[rowmax_]:=Table[Flatten[Table[ConstantArray[(n-m)(n-m+1)/2, PartitionsP[m]-PartitionsP[m-1]], {m, 0, n-1}]], {n, rowmax}];
A359279[10] (* Generates 10 rows *) (* Paolo Xausa, Mar 06 2023 *)

Prog

(PARI)
A359279(rowmax)=vector(rowmax, n, concat(vector(n, m, vector(numbpart(m-1)-numbpart(m-2), i, (n-m+1)*(n-m+2)/2))));
A359279(10) \\ Generates 10 rows - Paolo Xausa, Mar 06 2023

Crossrefs

Row sums give A014153 (convolution of A000041 and A000027).

This sequence has the same row sums as A176206, A299779 and A359350.

Cf. A000217, A336811, A336812.

Sequence in context: A108591 A110119 A322310 * A152202 A210039 A026250

Adjacent sequences: A359276 A359277 A359278 * A359280 A359281 A359282

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 23 2022

Status

approved