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A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

%I A340524 #15 Jan 13 2021 14:29:08
%S A340524 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 A340524 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 A340524 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 A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.
%C A340524 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 A340524 Triangle begins:
%e A340524 1;
%e A340524 2, 0;
%e A340524 2, 0, 1;
%e A340524 3, 0, 2, 1;
%e A340524 2, 0, 2, 2, 2;
%e A340524 4, 0, 3, 2, 4, 2;
%e A340524 2, 0, 2, 3, 4, 4,  4;
%e A340524 4, 0, 4, 2, 6, 4,  8,  4;
%e A340524 3, 0, 2, 4, 4, 6,  8,  8,  7;
%e A340524 4, 0, 4, 2, 8, 4, 12,  8, 14,  8;
%e A340524 2, 0, 3, 4, 4, 8,  8, 12, 14, 16, 12;
%e A340524 6, 0, 4, 3, 8, 4, 16,  8, 21, 16, 24, 14;
%e A340524 2, 0, 2, 4, 6, 8,  8, 16, 14, 24, 24, 28, 21;
%e A340524 ...
%e A340524 For n = 6 the calculation of every term of row 6 is as follows:
%e A340524 --------------------------
%e A340524 k   A002865         T(6,k)
%e A340524 --------------------------
%e A340524 1      1   *   4   =   4
%e A340524 2      0   *   2   =   0
%e A340524 3      1   *   3   =   3
%e A340524 4      1   *   2   =   2
%e A340524 5      2   *   2   =   4
%e A340524 6      2   *   1   =   2
%e A340524 .           A000005
%e A340524 --------------------------
%e A340524 The sum of row 6 is 4 + 0 + 3 + 2 + 4 + 2 = 15, equaling A138137(6) = 15.
%o A340524 (PARI) f(n) = if (n==0, 1, numbpart(n) - numbpart(n-1)); \\ A002865
%o A340524 T(n, k) = numdiv(n-k+1) * f(k-1); \\ _Michel Marcus_, Jan 13 2021
%Y A340524 Row sums give A138137 (conjectured).
%Y A340524 Columns 1, 3 and 4 are A000005.
%Y A340524 Column 2 gives A000004.
%Y A340524 Columns 5 and 6 give A062011.
%Y A340524 Columns 7 and 8 give A145154, n >= 1.
%Y A340524 Leading diagonal gives A002865.
%Y A340524 Cf. A339304 (irregular or expanded version).
%Y A340524 Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340426.
%K A340524 nonn,tabl
%O A340524 1,2
%A A340524 _Omar E. Pol_, Jan 10 2021