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A297321 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.

%I A297321 #16 Sep 22 2018 02:01:17
%S A297321 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 A297321 48,69,64,25,0,1,7,27,75,137,174,133,43,0,1,8,35,110,240,380,413,266,
%U A297321 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 A297321 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 A297321 G. C. Greubel, <a href="/A297321/b297321.txt">Table of n, a(n) for the first 100 antidiagonals, flattened</a>
%F A297321 G.f. of column k: Product_{j>=1} (1 + j*x^j)^k.
%e A297321 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 A297321 Square array begins:
%e A297321 1,   1,   1,    1,    1,    1,  ...
%e A297321 0,   1,   2,    3,    4,    5,  ...
%e A297321 0,   2,   5,    9,   14,   20,  ...
%e A297321 0,   5,  14,   28,   48,   75,  ...
%e A297321 0,   7,  28,   69,  137,  240,  ...
%e A297321 0,  15,  64,  174,  380,  726,  ...
%t A297321 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 A297321 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 A297321 Main diagonal gives A297322.
%Y A297321 Antidiagonal sums give A299164.
%Y A297321 Cf. A266891, A297323, A297325, A297328.
%K A297321 nonn,tabl
%O A297321 0,8
%A A297321 _Ilya Gutkovskiy_, Dec 28 2017