A292477 - OEIS

A292477

Square array A(n,k), n >= 0, k >= 2, read by antidiagonals: A(n,k) = [x^(k*n)] Product_{j>=0} 1/(1 - x^(k^j)).

%I #20 Sep 27 2020 12:30:14

%S 1,1,2,1,2,4,1,2,3,6,1,2,3,5,10,1,2,3,4,7,14,1,2,3,4,6,9,20,1,2,3,4,5,

%T 8,12,26,1,2,3,4,5,7,10,15,36,1,2,3,4,5,6,9,12,18,46,1,2,3,4,5,6,8,11,

%U 15,23,60,1,2,3,4,5,6,7,10,13,18,28,74,1,2,3,4,5,6,7,9,12,15,21,33,94

%N Square array A(n,k), n >= 0, k >= 2, read by antidiagonals: A(n,k) = [x^(k*n)] Product_{j>=0} 1/(1 - x^(k^j)).

%C A(n,k) is the number of partitions of k*n into powers of k.

%H Seiichi Manyama, <a href="/A292477/b292477.txt">Antidiagonals n = 0..139, flattened</a>

%H George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, <a href="https://doi.org/10.4169/amer.math.monthly.122.9.880">Characterizing the number of m-ary partitions modulo m</a>, Amer. Math. Monthly 122:9 (2015), 880-885.

%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>

%e Square array begins:

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

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

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

%e 6, 5, 4, 4, 4, 4, ...

%e 10, 7, 6, 5, 5, 5, ...

%e 14, 9, 8, 7, 6, 6, ...

%t Table[Function[k, SeriesCoefficient[Product[1/(1 - x^k^i), {i, 0, n}], {x, 0, k n}]][j - n + 2], {j, 0, 12}, {n, 0, j}] // Flatten

%Y Columns k=2..5 give A000123, A005704, A005705, A005706.

%Y Mirror of A089688 (excluding the first row).

%Y Cf. A145515, A294316.

%K nonn,tabl

%O 0,3

%A _Ilya Gutkovskiy_, Sep 17 2017