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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.

%I A207031 #107 Dec 09 2019 07:27:26
%S A207031 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,
%T A207031 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,
%U A207031 7,5,3,2,1,1,124,68,41,27,17,12,7,5,3,2,1,1
%N 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.
%C A207031 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.
%C A207031 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.
%C A207031 It appears that reversed rows converge to A000041.
%C A207031 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
%F A207031 From _Omar E. Pol_, Dec 07 2019: (Start)
%F A207031 From the formula in A138135 (year 2008) we have that:
%F A207031 A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
%F A207031 Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
%F A207031 Also  A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)
%e A207031 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):
%e A207031 .                                         6
%e A207031 .                                         3 3
%e A207031 .                                         4 2
%e A207031 .                                         2 2 2
%e A207031 .                            5              1
%e A207031 .                            3 2              1
%e A207031 .                  4           1              1
%e A207031 .                  2 2           1              1
%e A207031 .          3         1           1              1
%e A207031 .     2      1         1           1              1
%e A207031 .  1    1      1         1           1              1
%e A207031 . --- --- ------- --------- ----------- --------------
%e A207031 A: 1, 2,1, 3,1,1,  6,3,1,1,  8,3,2,1,1,  15,8,4,2,1,1
%e A207031 .  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|    |/|/|/|/|/|
%e A207031 B: 1, 1,1, 2,0,1,  3,2,0,1,  5,1,1,0,1,   7,4,2,1,0,1
%e A207031 .
%e A207031 A := initial terms of this triangle.
%e A207031 B := initial terms of triangle A182703.
%e A207031 .
%e A207031 Triangle begins:
%e A207031 1;
%e A207031 2,    1;
%e A207031 3,    1,  1;
%e A207031 6,    3,  1,  1;
%e A207031 8,    3,  2,  1,  1;
%e A207031 15,   8,  4,  2,  1,  1;
%e A207031 19,   8,  5,  3,  2,  1,  1;
%e A207031 32,  17,  9,  6,  3,  2,  1,  1;
%e A207031 42,  20, 13,  7,  5,  3,  2,  1,  1;
%e A207031 64,  34, 19, 13,  8,  5,  3,  2,  1,  1;
%e A207031 83,  41, 26, 16, 11,  7,  5,  3,  2,  1,  1;
%e A207031 124, 68, 41, 27, 17, 12,  7,  5,  3,  2,  1,  1;
%Y A207031 Columns (1-2): A138137, A138135.
%Y A207031 Row sums give A138879.
%Y A207031 Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.
%K A207031 nonn,tabl
%O A207031 1,2
%A A207031 _Omar E. Pol_, Feb 14 2012
%E A207031 More terms from _Alois P. Heinz_, Feb 17 2012