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%I #16 Sep 22 2018 02:01:17
%S 1,1,0,1,1,0,1,2,2,0,1,3,5,5,0,1,4,9,14,7,0,1,5,14,28,28,15,0,1,6,20,
%T 48,69,64,25,0,1,7,27,75,137,174,133,43,0,1,8,35,110,240,380,413,266,
%U 64,0,1,9,44,154,387,726,998,933,513,120,0,1,10,54,208,588,1266,2075,2488,2046,1000,186,0
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j*x^j)^k.
%H G. C. Greubel, <a href="/A297321/b297321.txt">Table of n, a(n) for the first 100 antidiagonals, flattened</a>
%F G.f. of column k: Product_{j>=1} (1 + j*x^j)^k.
%e G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 3)*x^2 + (1/6)*k*(k^2 + 9*k + 20)*x^3 + (1/24)*k*(k^3 + 18*k^2 + 107*k + 42)*x^4 + (1/120)*k*(k^4 + 30*k^3 + 335*k^2 + 810*k + 624)*x^5 + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 2, 5, 9, 14, 20, ...
%e 0, 5, 14, 28, 48, 75, ...
%e 0, 7, 28, 69, 137, 240, ...
%e 0, 15, 64, 174, 380, 726, ...
%t Table[Function[k, SeriesCoefficient[Product[(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y Columns k=0..32 give A000007, A022629, A022630, A022631, A022632, A022633, A022634, A022635, A022636, A022637, A022638, A022639, A022640, A022641, A022642, A022643, A022644, A022645, A022646, A022647, A022648, A022649, A022650, A022651, A022652, A022653, A022654, A022655, A022656, A022657, A022658, A022659, A022660.
%Y Main diagonal gives A297322.
%Y Antidiagonal sums give A299164.
%Y Cf. A266891, A297323, A297325, A297328.
%K nonn,tabl
%O 0,8
%A _Ilya Gutkovskiy_, Dec 28 2017
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