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

1,2

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)

Links

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

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).

Example

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;
  ...

Mathematica

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

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

Sequence in context: A091450 A163253 A191663 * A008831 A365378 A289506

Adjacent sequences: A206558 A206559 A206560 * A206562 A206563 A206564

Keyword

nonn,tabl

Author

Omar E. Pol, Feb 14 2012

Extensions

More terms from Alois P. Heinz, Feb 14 2012

Status

approved