cp's OEIS Frontend

A340525 Triangle read by rows: T(n,k) = A006218(n-k+1)*A002865(k-1), 1 <= k <= n.

%I A340525 #14 Jan 11 2021 23:10:43
%S A340525 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 A340525 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 A340525 29,0,23,20,32,28,40,32,35,24,12,35,0,27,23,40,32,56,40,56,40,36,14
%N A340525 Triangle read by rows: T(n,k) = A006218(n-k+1)*A002865(k-1), 1 <= k <= n.
%C A340525 Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.
%e A340525 Triangle begins:
%e A340525    1;
%e A340525    3,  0;
%e A340525    5,  0,  1;
%e A340525    8,  0,  3,  1;
%e A340525   10,  0,  5,  3,  2;
%e A340525   14,  0,  8,  5,  6,  2;
%e A340525   16,  0, 10,  8, 10,  6,  4;
%e A340525   20,  0, 14, 10, 16, 10, 12,  4;
%e A340525   23,  0, 16, 14, 20, 16, 20, 12,  7;
%e A340525   27,  0, 20, 16, 28, 20, 32, 20, 21,  8;
%e A340525   29,  0, 23, 20, 32, 28, 40, 32, 35, 24, 12;
%e A340525   35,  0, 27, 23, 40, 32, 56, 40, 56, 40, 36, 14;
%e A340525 ...
%e A340525 For n = 6 the calculation of every term of row 6 is as follows:
%e A340525 --------------------------
%e A340525 k   A002865         T(6,k)
%e A340525 --------------------------
%e A340525 1      1   *  14   =  14
%e A340525 2      0   *  10   =   0
%e A340525 3      1   *   8   =   8
%e A340525 4      1   *   5   =   5
%e A340525 5      2   *   3   =   6
%e A340525 6      2   *   1   =   2
%e A340525 .           A006218
%e A340525 --------------------------
%e A340525 The sum of row 6 is 14 + 0 + 8 + 5 + 6 + 2 = 35, equaling A006128(6).
%Y A340525 Mirror of A245095.
%Y A340525 Row sums give A006128 (conjectured).
%Y A340525 Columns 1, 3 and 4 are A006218.
%Y A340525 Column 2 gives A000004.
%Y A340525 Leading diagonal gives A002865.
%Y A340525 Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340524, A340426.
%K A340525 nonn,tabl
%O A340525 1,2
%A A340525 _Omar E. Pol_, Jan 10 2021