1,2

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

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)

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;

Columns (1-2): A138137, A138135.

Row sums give A138879.

Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.

Sequence in context: A350200 A129761 A319299 * A156248 A324817 A106406

Adjacent sequences: A207028 A207029 A207030 * A207032 A207033 A207034

nonn,tabl

Omar E. Pol, Feb 14 2012

More terms from Alois P. Heinz, Feb 17 2012

approved