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A329323 Triangle read by rows: T(n,k) is the sum of the parts congruent to 0 mod k in the partitions of n into equal parts, 1 <= k <= n.

%I A329323 #10 Nov 25 2019 01:04:32
%S A329323 1,4,2,6,0,3,12,8,0,4,10,0,0,0,5,24,12,12,0,0,6,14,0,0,0,0,0,7,32,24,
%T A329323 0,16,0,0,0,8,27,0,18,0,0,0,0,0,9,40,20,0,0,20,0,0,0,0,10,22,0,0,0,0,
%U A329323 0,0,0,0,0,11,72,48,36,24,0,24,0,0,0,0,0,12,26,0,0,0,0,0,0,0,0,0,0,0,13,56,28,0,0
%N A329323 Triangle read by rows: T(n,k) is the sum of the parts congruent to 0 mod k in the partitions of n into equal parts, 1 <= k <= n.
%C A329323 Column k lists the terms of A038040 multiplied by k and interspersed with (k-1) zeros.
%F A329323 T(n,k) = A126988(n,k)*A134577(n,k).
%e A329323 Triangle begins:
%e A329323    1;
%e A329323    4,  2;
%e A329323    6,  0,  3;
%e A329323   12,  8,  0,  4;
%e A329323   10,  0,  0,  0,  5;
%e A329323   24, 12, 12,  0,  0,  6;
%e A329323   14,  0,  0,  0,  0,  0,  7;
%e A329323   32, 24,  0, 16,  0,  0,  0,  8;
%e A329323   27,  0, 18,  0,  0,  0,  0,  0,  9;
%e A329323   40, 20,  0,  0, 20,  0,  0,  0,  0, 10;
%e A329323   22,  0,  0,  0,  0,  0,  0,  0,  0,  0, 11;
%e A329323   72, 48, 36, 24,  0, 24,  0,  0,  0,  0,  0, 12;
%e A329323   26,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 13;
%e A329323   56, 28,  0,  0,  0,  0, 28,  0,  0,  0,  0,  0,  0, 14;
%e A329323   60,  0, 30,  0, 30,  0,  0,  0,  0,  0,  0,  0,  0,  0, 15;
%e A329323   80, 64,  0, 48,  0,  0,  0, 32,  0,  0,  0,  0,  0,  0,  0, 16;
%e A329323 ...
%e A329323 For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. Then, for k = 2 the sum of the parts that are multiples of 2 is 6 + 2 + 2 + 2 = 12, so T(6,2) = 12.
%Y A329323 Column 1 is A038040.
%Y A329323 Row sums give A034718.
%Y A329323 Leading diagonal gives A000027.
%Y A329323 The number of positive terms in row n is A000005(n).
%Y A329323 Cf. A126988, A130540, A134577, A244051.
%K A329323 nonn,tabl
%O A329323 1,2
%A A329323 _Omar E. Pol_, Nov 21 2019