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

Author

Omar E. Pol, Dec 23 2022

Keywords

• nonn
• tabf

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.

Examples

			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)
		

Links

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

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.

Programs

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 *)

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

Formula

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