%I #45 Mar 21 2018 17:14:04

%S 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,

%T 13,7,176,131,93,66,42,27,15,8,270,203,151,106,74,49,31,17,9,420,323,

%U 241,178,126,86,56,35,19,10,616,477,365,272,200,140,98,63,39,21,11

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

%C From _Omar E. Pol_, Mar 18 2018: (Start)

%C 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):

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

%C --- ---- ------- ------------ ----------------

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

%C | | /| | /| /| | / | /| /| | / | / | /| /|

%C | |/ | |/ |/ | |/ |/ |/ | |/ |/ |/ |/ |

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

%H Alois P. Heinz, <a href="/A206561/b206561.txt">Rows n = 1..141, flattened</a>

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

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

%e Triangle begins:

%e 1;

%e 4, 2;

%e 9, 5, 3;

%e 20, 13, 7, 4;

%e 35, 23, 15, 9, 5;

%e 66, 47, 31, 19, 11, 6;

%e 105, 75, 53, 35, 23, 13, 7;

%e ...

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

%Y Columns 1-2 give A066186, A194552.

%Y Right border gives A000027.

%Y Cf. A138785, A181187.

%Y Row sums give A066183. - _Omar E. Pol_, Mar 19 2018

%Y Both A180681 and A299768 have the same row sums as this triangle. - _Omar E. Pol_, Mar 21 2018

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Feb 14 2012

%E More terms from _Alois P. Heinz_, Feb 14 2012