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A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

%I A286653 #26 Feb 16 2025 08:33:45
%S A286653 1,1,0,1,1,0,1,1,1,0,1,1,2,2,0,1,1,2,2,2,0,1,1,2,3,4,3,0,1,1,2,3,4,5,
%T A286653 4,0,1,1,2,3,5,6,7,5,0,1,1,2,3,5,6,9,9,6,0,1,1,2,3,5,7,10,12,13,8,0,1,
%U A286653 1,2,3,5,7,10,13,16,16,10,0,1,1,2,3,5,7,11,14,19,22,22,12,0
%N A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).
%C A286653 A(n,k) is the number of partitions of n in which no parts are multiples of k.
%C A286653 A(n,k) is also the number of partitions of n into at most k-1 copies of each part.
%H A286653 Seiichi Manyama, <a href="/A286653/b286653.txt">Antidiagonals n = 0..139, flattened</a>
%H A286653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function b_k</a>
%H A286653 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A286653 G.f. of column k: Product_{j>=1} (1 - x^(k*j))/(1 - x^j).
%e A286653 Square array begins:
%e A286653   1,  1,  1,  1,  1,  1,  ...
%e A286653   0,  1,  1,  1,  1,  1,  ...
%e A286653   0,  1,  2,  2,  2,  2,  ...
%e A286653   0,  2,  2,  3,  3,  3,  ...
%e A286653   0,  2,  4,  4,  5,  5,  ...
%e A286653   0,  3,  5,  6,  6,  7,  ...
%p A286653 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2<n, 0,
%p A286653       add((l->[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
%p A286653     end:
%p A286653 A:= (n, k)-> b(n$2, k-1)[1]:
%p A286653 seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # _Alois P. Heinz_, Oct 17 2018
%t A286653 Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
%t A286653 Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
%Y A286653 Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
%Y A286653 Main diagonal gives A000041.
%Y A286653 Mirror of A061198.
%Y A286653 Cf. A061199, A210485.
%K A286653 nonn,tabl
%O A286653 0,13
%A A286653 _Ilya Gutkovskiy_, May 11 2017