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A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

%I A340426 #26 Mar 07 2021 21:02:14
%S A340426 1,3,0,4,0,1,7,0,3,1,6,0,4,3,2,12,0,7,4,6,2,8,0,6,7,8,6,4,15,0,12,6,
%T A340426 14,8,12,4,13,0,8,12,12,14,16,12,7,18,0,15,8,24,12,28,16,21,8,12,0,13,
%U A340426 15,16,24,14,28,28,24,12,28,0,18,13,30,16,48,24,49,32,36,14,14,0,12
%N A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.
%C A340426 Conjecture: the sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.
%e A340426 Triangle begins:
%e A340426    1;
%e A340426    3,  0;
%e A340426    4,  0,  1;
%e A340426    7,  0,  3,  1;
%e A340426    6,  0,  4,  3,  2;
%e A340426   12,  0,  7,  4,  6,  2;
%e A340426    8,  0,  6,  7,  8,  6,  4;
%e A340426   15,  0, 12,  6, 14,  8, 12,  4;
%e A340426   13,  0,  8, 12, 12, 14, 16, 12,  7;
%e A340426   18,  0, 15,  8, 24, 12, 28, 16, 21,  8;
%e A340426   12,  0, 13, 15, 16, 24, 14, 28, 28, 24, 12;
%e A340426   28,  0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14;
%e A340426 ...
%e A340426 For n = 6 the calculation of every term of row 6 is as follows:
%e A340426 --------------------------
%e A340426 k   A002865         T(6,k)
%e A340426 --------------------------
%e A340426 1      1   *   12  =  12
%e A340426 2      0   *   6   =   0
%e A340426 3      1   *   7   =   7
%e A340426 4      1   *   4   =   4
%e A340426 5      2   *   3   =   6
%e A340426 6      2   *   1   =   2
%e A340426 .           A000203
%e A340426 --------------------------
%e A340426 The sum of row 6 is 12 + 0 + 7 + 4 + 6 + 2 = 31, equaling A138879(6) = 31.
%Y A340426 Columns 1, 3 and 4 give A000203.
%Y A340426 Column 2 gives A000004.
%Y A340426 Columns 5 and 6 gives A074400.
%Y A340426 Column 7 and 8 give A239050.
%Y A340426 Column 9 gives A319527.
%Y A340426 Column 10 gives A319528.
%Y A340426 Leading diagonal gives A002865.
%Y A340426 Cf. A135010, A138879.
%K A340426 nonn,tabl
%O A340426 1,2
%A A340426 _Omar E. Pol_, Jan 07 2021