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

%I A206561 #45 Mar 21 2018 17:14:04
%S A206561 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 A206561 13,7,176,131,93,66,42,27,15,8,270,203,151,106,74,49,31,17,9,420,323,
%U A206561 241,178,126,86,56,35,19,10,616,477,365,272,200,140,98,63,39,21,11
%N A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.
%C A206561 From _Omar E. Pol_, Mar 18 2018: (Start)
%C A206561 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 A206561 Row..........:  1     2       3            4               5          ...
%C A206561                ---  ----   -------   ------------   ----------------
%C A206561 This triangle:  1;  4, 2;  9, 5, 3;  20, 13, 7, 4;  35, 23, 15, 9, 5; ...
%C A206561                 |   | /|   | /| /|    | / | /| /|    | / | / | /| /|
%C A206561                 |   |/ |   |/ |/ |    |/  |/ |/ |    |/  |/  |/ |/ |
%C A206561 A138785......:  1;  2, 2;  4, 2, 3;   7,  6, 3, 4;  12,  8,  6, 4, 5; ... (End)
%H A206561 Alois P. Heinz, <a href="/A206561/b206561.txt">Rows n = 1..141, flattened</a>
%F A206561 T(n,n) = n, T(n,k) = T(n,k+1) + k * A066633(n,k) for k < n.
%F A206561 T(n,k) = Sum_{i=k..n} A138785(n,i).
%e A206561 Triangle begins:
%e A206561     1;
%e A206561     4,  2;
%e A206561     9,  5,  3;
%e A206561    20, 13,  7,  4;
%e A206561    35, 23, 15,  9,  5;
%e A206561    66, 47, 31, 19, 11,  6;
%e A206561   105, 75, 53, 35, 23, 13,  7;
%e A206561   ...
%t A206561 Table[With[{s = IntegerPartitions[n]}, Table[Total@ Flatten@ Map[Select[#, # >= k &] &, s], {k, n}]], {n, 11}] // Flatten (* _Michael De Vlieger_, Mar 19 2018 *)
%Y A206561 Columns 1-2 give A066186, A194552.
%Y A206561 Right border gives A000027.
%Y A206561 Cf. A138785, A181187.
%Y A206561 Row sums give A066183. - _Omar E. Pol_, Mar 19 2018
%Y A206561 Both A180681 and A299768 have the same row sums as this triangle. - _Omar E. Pol_, Mar 21 2018
%K A206561 nonn,tabl
%O A206561 1,2
%A A206561 _Omar E. Pol_, Feb 14 2012
%E A206561 More terms from _Alois P. Heinz_, Feb 14 2012