|
%I #14 Jan 11 2021 23:10:43
%S 1,3,0,5,0,1,8,0,3,1,10,0,5,3,2,14,0,8,5,6,2,16,0,10,8,10,6,4,20,0,14,
%T 10,16,10,12,4,23,0,16,14,20,16,20,12,7,27,0,20,16,28,20,32,20,21,8,
%U 29,0,23,20,32,28,40,32,35,24,12,35,0,27,23,40,32,56,40,56,40,36,14
%N Triangle read by rows: T(n,k) = A006218(n-k+1)*A002865(k-1), 1 <= k <= n.
%C Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.
%e Triangle begins:
%e 1;
%e 3, 0;
%e 5, 0, 1;
%e 8, 0, 3, 1;
%e 10, 0, 5, 3, 2;
%e 14, 0, 8, 5, 6, 2;
%e 16, 0, 10, 8, 10, 6, 4;
%e 20, 0, 14, 10, 16, 10, 12, 4;
%e 23, 0, 16, 14, 20, 16, 20, 12, 7;
%e 27, 0, 20, 16, 28, 20, 32, 20, 21, 8;
%e 29, 0, 23, 20, 32, 28, 40, 32, 35, 24, 12;
%e 35, 0, 27, 23, 40, 32, 56, 40, 56, 40, 36, 14;
%e ...
%e For n = 6 the calculation of every term of row 6 is as follows:
%e --------------------------
%e k A002865 T(6,k)
%e --------------------------
%e 1 1 * 14 = 14
%e 2 0 * 10 = 0
%e 3 1 * 8 = 8
%e 4 1 * 5 = 5
%e 5 2 * 3 = 6
%e 6 2 * 1 = 2
%e . A006218
%e --------------------------
%e The sum of row 6 is 14 + 0 + 8 + 5 + 6 + 2 = 35, equaling A006128(6).
%Y Mirror of A245095.
%Y Row sums give A006128 (conjectured).
%Y Columns 1, 3 and 4 are A006218.
%Y Column 2 gives A000004.
%Y Leading diagonal gives A002865.
%Y Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340524, A340426.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Jan 10 2021
|
