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%I #4 Oct 08 2018 18:15:23
%S 1,1,1,1,1,2,1,1,3,4,1,1,5,8,8,1,1,9,18,21,16,1,1,17,44,63,55,32,1,1,
%T 33,114,207,221,144,64,1,1,65,308,723,991,776,377,128,1,1,129,858,
%U 2631,4805,4752,2725,987,256,1,1,257,2444,9843,24655,31880,22769,9569,2584,512
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).
%C A(n,k) is the invert transform of k-th powers evaluated at n.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f. of column k: 1/(1 - PolyLog(-k,x)), where PolyLog() is the polylogarithm function.
%e G.f. of column k: A_k(x) = 1 + x + (2^k + 1)*x^2 + (2^(k + 1) + 3^k + 1)*x^3 + (3*2^k + 2^(2*k + 1) + 2*3^k + 1)*x^4 + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 2, 3, 5, 9, 17, 33, ...
%e 4, 8, 18, 44, 114, 308, ...
%e 8, 21, 63, 207, 723, 2631, ...
%e 16, 55, 221, 991, 4805, 24655, ...
%t Table[Function[k, SeriesCoefficient[1/(1 - Sum[i^k x^i, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t Table[Function[k, SeriesCoefficient[1/(1 - PolyLog[-k, x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y Columns k=0..3 give A011782, A088305, A033453, A144109.
%Y Main diagonal gives A301655.
%Y Cf. A144048.
%K nonn,tabl
%O 0,6
%A _Ilya Gutkovskiy_, Oct 08 2018