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A207031
Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.
32
1, 2, 1, 3, 1, 1, 6, 3, 1, 1, 8, 3, 2, 1, 1, 15, 8, 4, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 32, 17, 9, 6, 3, 2, 1, 1, 42, 20, 13, 7, 5, 3, 2, 1, 1, 64, 34, 19, 13, 8, 5, 3, 2, 1, 1, 83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1, 124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1
OFFSET
1,2
COMMENTS
Also T(n,k) is the number of parts >= k in the last section of the set of partitions of n. Therefore T(n,1) = A138137(n), the total number of parts in the last section of the set of partitions of n. For calculation of the number of odd/even parts, etc, follow the same rules from A206563.
More generally, let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the entry A206563.
It appears that reversed rows converge to A000041.
It appears that the first differences of row n together with 1 give the row n of triangle A182703 (see example). - Omar E. Pol, Feb 26 2012
FORMULA
From Omar E. Pol, Dec 07 2019: (Start)
From the formula in A138135 (year 2008) we have that:
A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
Also A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)
EXAMPLE
Illustration of initial terms. First six rows of triangle as sums of columns from the last sections of the first six natural numbers (or as sums of columns from the six sections of 6):
. 6
. 3 3
. 4 2
. 2 2 2
. 5 1
. 3 2 1
. 4 1 1
. 2 2 1 1
. 3 1 1 1
. 2 1 1 1 1
. 1 1 1 1 1 1
. --- --- ------- --------- ----------- --------------
A: 1, 2,1, 3,1,1, 6,3,1,1, 8,3,2,1,1, 15,8,4,2,1,1
. | |/| |/|/| |/|/|/| |/|/|/|/| |/|/|/|/|/|
B: 1, 1,1, 2,0,1, 3,2,0,1, 5,1,1,0,1, 7,4,2,1,0,1
.
A := initial terms of this triangle.
B := initial terms of triangle A182703.
.
Triangle begins:
1;
2, 1;
3, 1, 1;
6, 3, 1, 1;
8, 3, 2, 1, 1;
15, 8, 4, 2, 1, 1;
19, 8, 5, 3, 2, 1, 1;
32, 17, 9, 6, 3, 2, 1, 1;
42, 20, 13, 7, 5, 3, 2, 1, 1;
64, 34, 19, 13, 8, 5, 3, 2, 1, 1;
83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1;
124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1;
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Feb 14 2012
EXTENSIONS
More terms from Alois P. Heinz, Feb 17 2012
STATUS
approved