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Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.
5

%I #25 Dec 17 2018 18:48:46

%S 1,3,1,5,3,1,9,7,3,1,12,12,7,3,1,20,21,14,7,3,1,25,31,24,14,7,3,1,38,

%T 47,40,26,14,7,3,1,49,66,61,43,26,14,7,3,1,69,93,92,70,45,26,14,7,3,1,

%U 87,124,130,106,73,45,26,14,7,3,1

%N Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.

%C In row n, the sum of all odd-indexed terms minus the sum of all even-indexed terms is equal to A194714(n).

%C Reversed rows converge to A014153. - _Alois P. Heinz_, Feb 13 2012

%H Alois P. Heinz, <a href="/A206283/b206283.txt">Rows n = 1..70, flattened</a>

%e Row 4 is 9, 7, 3, 1 because the five partitions of 4, with their parts written in nondecreasing order, are

%e . 4

%e . 1, 3

%e . 2, 2

%e . 1, 1, 2

%e . 1, 1, 1, 1

%e -------------------------------------------

%e And the sums of the columns are 9, 7, 3, 1.

%e .

%e Triangle begins:

%e 1;

%e 3, 1;

%e 5, 3, 1;

%e 9, 7, 3, 1;

%e 12, 12, 7, 3, 1;

%e 20, 21, 14, 7, 3, 1;

%e 25, 31, 24, 14, 7, 3, 1;

%e 38, 47, 40, 26, 14, 7, 3, 1;

%e 49, 66, 61, 43, 26, 14, 7, 3, 1;

%e 69, 93, 92, 70, 45, 26, 14, 7, 3, 1;

%Y Column 1 is A046746. Row sums give A066186.

%Y Cf. A014153, A181187, A194714.

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Feb 13 2012

%E More terms from _Alois P. Heinz_, Feb 13 2012