A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.

1, 4, 2, 9, 5, 3, 20, 13, 7, 4, 35, 23, 15, 9, 5, 66, 47, 31, 19, 11, 6, 105, 75, 53, 35, 23, 13, 7, 176, 131, 93, 66, 42, 27, 15, 8, 270, 203, 151, 106, 74, 49, 31, 17, 9, 420, 323, 241, 178, 126, 86, 56, 35, 19, 10, 616, 477, 365, 272, 200, 140, 98, 63, 39, 21, 11

Offset: 1

Author

Omar E. Pol, Feb 14 2012

Keywords

• nonn
• tabl

Comments

From Omar E. Pol, Mar 18 2018: (Start)

In the n-th row of the triangle the first differences together with its last term give the n-th row of triangle A138785 (see below):

Row..........: 1 2 3 4 5 ...

--- ---- ------- ------------ ----------------

This triangle: 1; 4, 2; 9, 5, 3; 20, 13, 7, 4; 35, 23, 15, 9, 5; ...

| | /| | /| /| | / | /| /| | / | / | /| /|

| |/ | |/ |/ | |/ |/ |/ | |/ |/ |/ |/ |

A138785......: 1; 2, 2; 4, 2, 3; 7, 6, 3, 4; 12, 8, 6, 4, 5; ... (End)

Examples

			Triangle begins:
    1;
    4,  2;
    9,  5,  3;
   20, 13,  7,  4;
   35, 23, 15,  9,  5;
   66, 47, 31, 19, 11,  6;
  105, 75, 53, 35, 23, 13,  7;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Columns 1-2 give A066186, A194552.

Right border gives A000027.

Cf. A138785, A181187.

Row sums give A066183. - Omar E. Pol, Mar 19 2018

Both A180681 and A299768 have the same row sums as this triangle. - Omar E. Pol, Mar 21 2018

Programs

Mathematica

Table[With[{s = IntegerPartitions[n]}, Table[Total@ Flatten@ Map[Select[#, # >= k &] &, s], {k, n}]], {n, 11}] // Flatten (* Michael De Vlieger, Mar 19 2018 *)

Formula

T(n,n) = n, T(n,k) = T(n,k+1) + k * A066633(n,k) for k < n.

T(n,k) = Sum_{i=k..n} A138785(n,i).

Extensions

More terms from Alois P. Heinz, Feb 14 2012