A352499 Irregular triangle read by rows: T(n,k) is the sum of all parts of the partition of n into consecutive parts that contains 2*k-1 parts, and the first element of the column k is in row A000384(k).

1, 2, 3, 4, 5, 6, 6, 7, 0, 8, 0, 9, 9, 10, 0, 11, 0, 12, 12, 13, 0, 14, 0, 15, 15, 15, 16, 0, 0, 17, 0, 0, 18, 18, 0, 19, 0, 0, 20, 0, 20, 21, 21, 0, 22, 0, 0, 23, 0, 0, 24, 24, 0, 25, 0, 25, 26, 0, 0, 27, 27, 0, 28, 0, 0, 28, 29, 0, 0, 0, 30, 30, 30, 0, 31, 0, 0, 0, 32, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Mar 19 2022

Keywords

• nonn
• tabf

Comments

This triangle is formed from the odd-indexed columns of the triangle A285891.

Examples

			Triangle begins:
   1;
   2;
   3;
   4;
   5;
   6,  6;
   7,  0;
   8,  0;
   9,  9;
  10,  0;
  11,  0;
  12, 12;
  13,  0;
  14,  0;
  15, 15, 15;
  16,  0,  0;
  17,  0,  0;
  18, 18,  0;
  19,  0,  0;
  20,  0, 20;
  21, 21,  0;
  22,  0,  0;
  23,  0,  0;
  24, 24,  0;
  25,  0, 25;
  26,  0,  0;
  27, 27,  0;
  28,  0,  0, 28;
  ...
For n = 21 the partitions of 21 into on odd number of consecutive parts are [21] and [8, 7, 6], so T(21,1) = 1 and T(21,2) = 8 + 7 + 6 = 21. There is no partition of 21 into five consecutive parts so T(21,3) = 0.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10490 (rows 1..800 of triangle, flattened).

Crossrefs

Row sums give A352257.

Row n has A351846(n) terms.

The number of nonzero terms in row n equals A082647(n).

Cf. A000384, A237048, A245579, A285891, A299765, A351824, A352425.

Programs

Mathematica

A352499[rowmax_]:=Table[Boole[Divisible[n,2k-1]]n,{n,rowmax},{k,Floor[(Sqrt[8n+1]+1)/4]}];A352499[50] (* Paolo Xausa, Apr 12 2023 *)

Formula

T(n,k) = n*A351824(n,k).

T(n,k) = n*[(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - Paolo Xausa, Apr 12 2023