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Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Sum_{j=0..n} floor(n/k^j).
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%I #7 Feb 22 2019 05:17:17

%S 0,0,2,0,1,6,0,1,3,12,0,1,2,4,20,0,1,2,4,7,30,0,1,2,3,5,8,42,0,1,2,3,

%T 5,6,10,56,0,1,2,3,4,6,8,11,72,0,1,2,3,4,6,7,9,15,90,0,1,2,3,4,5,7,8,

%U 10,16,110,0,1,2,3,4,5,7,8,10,13,18,132,0,1,2,3,4,5,6,8,9,11,14,19,156

%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Sum_{j=0..n} floor(n/k^j).

%F G.f. of column k (for k > 1): (1/(1 - x)) * Sum_{j>=0} x^(k^j)/(1 - x^(k^j)).

%e Square array begins:

%e 0, 0, 0, 0, 0, 0, ...

%e 2, 1, 1, 1, 1, 1, ...

%e 6, 3, 2, 2, 2, 2, ...

%e 12, 4, 4, 3, 3, 3, ...

%e 20, 7, 5, 5, 4, 4, ...

%e 30, 8, 6, 6, 6, 5, ...

%t Table[Function[k, Sum[Floor[n/k^j], {j, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

%Y Columns k=1..4 give A002378, A005187, A004128, A087069.

%Y Cf. A306533.

%K nonn,tabl

%O 0,3

%A _Ilya Gutkovskiy_, Feb 22 2019