%I #15 Jan 13 2021 14:29:08
%S 1,2,0,2,0,1,3,0,2,1,2,0,2,2,2,4,0,3,2,4,2,2,0,2,3,4,4,4,4,0,4,2,6,4,
%T 8,4,3,0,2,4,4,6,8,8,7,4,0,4,2,8,4,12,8,14,8,2,0,3,4,4,8,8,12,14,16,
%U 12,6,0,4,3,8,4,16,8,21,16,24,14,2,0,2,4,6,8,8,16,14,24,24,28,21
%N Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.
%C Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.
%e Triangle begins:
%e 1;
%e 2, 0;
%e 2, 0, 1;
%e 3, 0, 2, 1;
%e 2, 0, 2, 2, 2;
%e 4, 0, 3, 2, 4, 2;
%e 2, 0, 2, 3, 4, 4, 4;
%e 4, 0, 4, 2, 6, 4, 8, 4;
%e 3, 0, 2, 4, 4, 6, 8, 8, 7;
%e 4, 0, 4, 2, 8, 4, 12, 8, 14, 8;
%e 2, 0, 3, 4, 4, 8, 8, 12, 14, 16, 12;
%e 6, 0, 4, 3, 8, 4, 16, 8, 21, 16, 24, 14;
%e 2, 0, 2, 4, 6, 8, 8, 16, 14, 24, 24, 28, 21;
%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 * 4 = 4
%e 2 0 * 2 = 0
%e 3 1 * 3 = 3
%e 4 1 * 2 = 2
%e 5 2 * 2 = 4
%e 6 2 * 1 = 2
%e . A000005
%e --------------------------
%e The sum of row 6 is 4 + 0 + 3 + 2 + 4 + 2 = 15, equaling A138137(6) = 15.
%o (PARI) f(n) = if (n==0, 1, numbpart(n) - numbpart(n-1)); \\ A002865
%o T(n, k) = numdiv(n-k+1) * f(k-1); \\ _Michel Marcus_, Jan 13 2021
%Y Row sums give A138137 (conjectured).
%Y Columns 1, 3 and 4 are A000005.
%Y Column 2 gives A000004.
%Y Columns 5 and 6 give A062011.
%Y Columns 7 and 8 give A145154, n >= 1.
%Y Leading diagonal gives A002865.
%Y Cf. A339304 (irregular or expanded version).
%Y Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340426.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Jan 10 2021