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A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

%I A206563 #98 Feb 23 2020 09:38:52
%S A206563 1,2,1,5,1,1,8,4,1,1,15,5,3,1,1,24,11,5,3,1,1,39,15,9,4,3,1,1,58,28,
%T A206563 13,9,4,3,1,1,90,38,23,12,8,4,3,1,1,130,62,33,21,12,8,4,3,1,1,190,85,
%U A206563 51,29,20,11,8,4,3,1,1,268,131,73,48,28,20,11,8,4,3,1,1
%N A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.
%C A206563 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 section example. (Cf. A135010, A207031, A207032, A212010). - _Omar E. Pol_, May 01 2012
%F A206563 It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A181187(n,j)).
%F A206563 It appears that A066633(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012
%e A206563 Calculation for n = 6. Write the partitions of 6 and below the sums of their columns:
%e A206563 .
%e A206563 .   6
%e A206563 .   3 + 3
%e A206563 .   4 + 2
%e A206563 .   2 + 2 + 2
%e A206563 .   5 + 1
%e A206563 .   3 + 2 + 1
%e A206563 .   4 + 1 + 1
%e A206563 .   2 + 2 + 1 + 1
%e A206563 .   3 + 1 + 1 + 1
%e A206563 .   2 + 1 + 1 + 1 + 1
%e A206563 .   1 + 1 + 1 + 1 + 1 + 1
%e A206563 . ------------------------
%e A206563 .  35, 16,  8,  4,  2,  1  --> Row 6 of triangle A181187.
%e A206563 .   |  /|  /|  /|  /|  /|
%e A206563 .   | / | / | / | / | / |
%e A206563 .   |/  |/  |/  |/  |/  |
%e A206563 .  19,  8,  4,  2,  1,  1  --> Row 6 of triangle A066633.
%e A206563 .
%e A206563 More generally, it appears that the sum of column k is also the total number of parts >= k in all partitions of n. It appears that the first differences of the column sums together with 1 give the number of occurrences of k in all partitions of n.
%e A206563 On the other hand we can see that the partitions of 6 contain:
%e A206563 24  odd parts >= 1 (the odd parts).
%e A206563 11 even parts >= 2 (the even parts).
%e A206563 5   odd parts >= 3.
%e A206563 3  even parts >= 4.
%e A206563 2   odd parts >= 5.
%e A206563 1  even part  >= 6.
%e A206563 Then, using the values of the column sums, it appears that:
%e A206563 T(6,1) = 35 - 16 + 8 - 4 + 2 - 1 = 24
%e A206563 T(6,2) =      16 - 8 + 4 - 2 + 1 = 11
%e A206563 T(6,3) =           8 - 4 + 2 - 1 = 5
%e A206563 T(6,4) =               4 - 2 + 1 = 3
%e A206563 T(6,5) =                   2 - 1 = 1
%e A206563 T(6,6) =                       1 = 1
%e A206563 So the 6th row of our triangle gives 24, 11, 5, 3, 1, 1.
%e A206563 Finally, for all partitions of 6, we can write:
%e A206563 The number of  odd parts      is equal to T(6,1) = 24.
%e A206563 The number of even parts      is equal to T(6,2) = 11.
%e A206563 The number of  odd parts >= 3 is equal to T(6,3) = 5.
%e A206563 The number of even parts >= 4 is equal to T(6,4) = 3.
%e A206563 The number of  odd parts >= 5 is equal to T(6,5) = 1.
%e A206563 The number of even parts >= 6 is equal to T(6,6) = 1.
%e A206563 More generally, we can write the same properties for any positive integer.
%e A206563 Triangle begins:
%e A206563 1;
%e A206563 2,    1;
%e A206563 5,    1,  1;
%e A206563 8,    4,  1,  1;
%e A206563 15,   5,  3,  1,  1;
%e A206563 24,  11,  5,  3,  1,  1;
%e A206563 39,  15,  9,  4,  3,  1,  1;
%e A206563 58,  28, 13,  9,  4,  3,  1,  1;
%e A206563 90,  38, 23, 12,  8,  4,  3,  1,  1;
%e A206563 130, 62, 33, 21, 12,  8,  4,  3,  1,  1;
%Y A206563 Columns 1-2 give A066897, A066898.
%Y A206563 Cf. A006128, A066633, A181187, A182703, A207031, A207032.
%K A206563 nonn,tabl
%O A206563 1,2
%A A206563 _Omar E. Pol_, Feb 15 2012
%E A206563 More terms from _Alois P. Heinz_, Feb 18 2012