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Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).
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%I #7 Feb 21 2019 04:17:22

%S 1,1,1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,2,2,5,1,1,1,1,3,3,8,1,1,1,2,1,6,4,

%T 13,1,1,1,1,4,1,10,6,21,1,1,1,2,1,7,2,18,9,34,1,1,1,1,3,1,13,3,31,13,

%U 55,1,1,1,2,2,6,1,25,4,55,19,89,1,1,1,1,3,3,10,1,46,5,96,28,144,1

%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).

%C A(n,k) is the number of compositions (ordered partitions) of n into divisors of k.

%F G.f. of column k: 1/(1 - Sum_{d|k} x^d).

%e Square array begins:

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

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

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

%e 1, 3, 2, 3, 1, 4, ...

%e 1, 5, 3, 6, 1, 7, ...

%e 1, 8, 4, 10, 2, 13, ...

%t Table[Function[k, SeriesCoefficient[1/(1 - Sum[x^d, {d, Divisors[k]}]), {x, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

%Y Columns k=1..7 give A000012, A000045 (for n > 0), A000930, A060945, A003520, A079958, A005709.

%Y Cf. A100346, A214575.

%K nonn,tabl

%O 0,9

%A _Ilya Gutkovskiy_, Feb 19 2019