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A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

%I A207383 #32 Nov 29 2020 12:38:59
%S A207383 1,1,2,2,0,3,3,4,0,4,5,2,3,0,5,7,8,6,4,0,6,11,6,6,4,5,0,7,15,16,9,12,
%T A207383 5,6,0,8,22,14,18,8,10,6,7,0,9,30,30,18,20,15,12,7,8,0,10,42,30,30,20,
%U A207383 20,12,14,8,9,0,11,56,54,42,40,25,30,14,16,9,10,0,12
%N A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.
%C A207383 For further properties of this triangle see also A182703.
%H A207383 Alois P. Heinz, <a href="/A207383/b207383.txt">Rows n = 1..141, flattened</a>
%F A207383 T(n,k) = k*A182703(n,k).
%e A207383 Triangle begins:
%e A207383    1;
%e A207383    1,  2;
%e A207383    2,  0,  3;
%e A207383    3,  4,  0,  4;
%e A207383    5,  2,  3,  0,  5;
%e A207383    7,  8,  6,  4,  0,  6;
%e A207383   11,  6,  6,  4,  5,  0,  7;
%e A207383   15, 16,  9, 12,  5,  6,  0,  8;
%e A207383   22, 14, 18,  8, 10,  6,  7,  0,  9;
%e A207383   30, 30, 18, 20, 15, 12,  7,  8,  0, 10;
%e A207383   42, 30, 30, 20, 20, 12, 14,  8,  9,  0, 11;
%e A207383   56, 54, 42, 40, 25, 30, 14, 16,  9, 10,  0, 12;
%e A207383 ...
%e A207383 From _Omar E. Pol_, Nov 28 2020: (Start)
%e A207383 Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
%e A207383 .                                        _ _ _ _ _ _ _
%e A207383 .     (7)                    (7)        |_ _ _ _      |
%e A207383 .     (4+3)                (4+3)        |_ _ _ _|_    |
%e A207383 .     (5+2)                (5+2)        |_ _ _    |   |
%e A207383 .     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
%e A207383 .       (1)                  (1)                    | |
%e A207383 .         (1)                (1)                    | |
%e A207383 .         (1)                (1)                    | |
%e A207383 .           (1)              (1)                    | |
%e A207383 .         (1)                (1)                    | |
%e A207383 .           (1)              (1)                    | |
%e A207383 .           (1)              (1)                    | |
%e A207383 .             (1)            (1)                    | |
%e A207383 .             (1)            (1)                    | |
%e A207383 .               (1)          (1)                    | |
%e A207383 .                 (1)        (1)                    |_|
%e A207383 .    ----------------
%e A207383 .     19,8,5,3,2,1,1 --> Row 7 of triangle A207031
%e A207383 .      |/|/|/|/|/|/|
%e A207383 .     11,3,2,1,1,0,1 --> Row 7 of triangle A182703
%e A207383 .      * * * * * * *
%e A207383 .      1,2,3,4,5,6,7 --> Row 7 of triangle A002260
%e A207383 .      = = = = = = =
%e A207383 .     11,6,6,4,5,0,7 --> Row 7 of this triangle
%e A207383 .
%e A207383 Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So the 7th row of triangle is [11, 6, 6, 4, 5, 0, 7]. (End)
%Y A207383 Column 1 is A000041.
%Y A207383 Leading diagonal gives A000027.
%Y A207383 Second diagonal gives A000007.
%Y A207383 Row sums give A138879.
%Y A207383 Cf. A002260, A066186, A135010, A138121, A138785, A138880, A182703, A194812, A207031.
%K A207383 nonn,tabl
%O A207383 1,3
%A A207383 _Omar E. Pol_, Feb 24 2012