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%I #11 Sep 09 2017 07:05:34
%S 1,1,0,1,1,0,1,2,1,0,1,3,2,2,0,1,4,3,6,2,0,1,5,4,12,6,3,0,1,6,5,20,12,
%T 10,4,0,1,7,6,30,20,21,18,5,0,1,8,7,42,30,36,48,22,6,0,1,9,8,56,42,55,
%U 100,57,30,8,0,1,10,9,72,56,78,180,116,84,42,10,0,1,11,10,90,72,105,294,205,180,120,66,12,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).
%C A(n,k) is the number of partitions of n into distinct parts of k sorts: the parts are unordered, but not the sorts.
%H Seiichi Manyama, <a href="/A286957/b286957.txt">Antidiagonals n = 0..139, flattened</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f. of column k: Product_{j>=1} (1 + k*x^j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 2, 6, 12, 20, 30, ...
%e 0, 2, 6, 12, 20, 30, ...
%e 0, 3, 10, 21, 36, 55, ...
%t Table[Function[k, SeriesCoefficient[Product[(1 + k x^i), {i, 1, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%t Table[Function[k, SeriesCoefficient[QPochhammer[-k, x]/(1 + k), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%Y Columns k=0-5 give: A000007, A000009, A032302, A032308, A261568, A261569.
%Y Main diagonal gives A291698.
%Y Cf. A246935.
%K nonn,tabl
%O 0,8
%A _Ilya Gutkovskiy_, May 17 2017
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