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Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.
6

%I #33 Jan 10 2021 11:23:12

%S 1,4,0,8,0,1,15,0,4,1,21,0,8,4,2,33,0,15,8,8,2,41,0,21,15,16,8,4,56,0,

%T 33,21,30,16,16,4,69,0,41,33,42,30,32,16,7,87,0,56,41,66,42,60,32,28,

%U 8,99,0,69,56,82,66,84,60,56,32,12,127,0,87,69,112,82,132,84,105,64,48,14

%N Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

%C Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.

%e Triangle begins:

%e 1;

%e 4, 0;

%e 8, 0, 1;

%e 15, 0, 4, 1;

%e 21, 0, 8, 4, 2;

%e 33, 0, 15, 8, 8, 2;

%e 41, 0, 21, 15, 16 8, 4;

%e 56, 0, 33, 21, 30, 16, 16, 4;

%e 69, 0, 41, 33, 42, 30, 32, 16, 7;

%e 87, 0, 56, 41, 66, 42, 60, 32, 28, 8;

%e 99, 0, 69, 56, 82, 66, 84, 60, 56, 32, 12;

%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 * 33 = 33

%e 2 0 * 21 = 0

%e 3 1 * 15 = 15

%e 4 1 * 8 = 8

%e 5 2 * 4 = 8

%e 6 2 * 1 = 2

%e . A024916

%e --------------------------

%e The sum of row 6 is 33 + 0 + 15 + 8 + 8 + 2 = 66, equaling A066186(6) = 66.

%Y Mirror of A245099.

%Y Columns 1, 3 and 4 are A024916 (partial sums of A000203).

%Y Column 2 gives A000004.

%Y Columns 5 and 6 give A327329.

%Y Columns 7 and 8 give A243980.

%Y Leading diagonal gives A002865.

%Y Cf. A066186.

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Jan 07 2021