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%I #5 Mar 29 2019 15:51:20
%S 1,1,0,1,1,0,1,2,3,0,1,3,16,21,0,1,4,45,416,315,0,1,5,96,2835,33280,
%T 9765,0,1,6,175,11904,722925,8053760,615195,0,1,7,288,37625,7428096,
%U 739552275,5863137280,78129765,0,1,8,441,98496,48724375,23205371904,3028466566125,12816818094080,19923090075,0
%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Product_{j=1..n} (k^j - 1).
%F G.f. of column k: Sum_{i>=0} k^(i*(i+1)/2)*x^i / Product_{j=0..i} (1 + k^j*x).
%F For asymptotics of column k see comment from _Vaclav Kotesovec_ in A027880.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 3, 16, 45, 96, 175, ...
%e 0, 21, 416, 2835, 11904, 37625, ...
%e 0, 315, 33280, 722925, 7428096, 48724375, ...
%e 0, 9765, 8053760, 739552275, 23205371904, 378832015625, ...
%t Table[Function[k, Product[k^j - 1, {j, 1, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
%t Table[Function[k, SeriesCoefficient[Sum[k^(i (i + 1)/2) x^i/Product[(1 + k^j x), {j, 0, i}], {i, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
%Y Columns k=1..12 give A000007, A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880.
%Y Cf. A069777, A225816.
%K nonn,tabl
%O 0,8
%A _Ilya Gutkovskiy_, Oct 11 2018
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