cp's OEIS Frontend

A320251 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).

%I A320251 #4 Oct 08 2018 18:15:23
%S A320251 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 A320251 33,114,207,221,144,64,1,1,65,308,723,991,776,377,128,1,1,129,858,
%U A320251 2631,4805,4752,2725,987,256,1,1,257,2444,9843,24655,31880,22769,9569,2584,512
%N A320251 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 A320251 A(n,k) is the invert transform of k-th powers evaluated at n.
%H A320251 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A320251 G.f. of column k: 1/(1 - PolyLog(-k,x)), where PolyLog() is the polylogarithm function.
%e A320251 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 A320251 Square array begins:
%e A320251    1,   1,    1,    1,     1,      1,  ...
%e A320251    1,   1,    1,    1,     1,      1,  ...
%e A320251    2,   3,    5,    9,    17,     33,  ...
%e A320251    4,   8,   18,   44,   114,    308,  ...
%e A320251    8,  21,   63,  207,   723,   2631,  ...
%e A320251   16,  55,  221,  991,  4805,  24655,  ...
%t A320251 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 A320251 Table[Function[k, SeriesCoefficient[1/(1 - PolyLog[-k, x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y A320251 Columns k=0..3 give A011782, A088305, A033453, A144109.
%Y A320251 Main diagonal gives A301655.
%Y A320251 Cf. A144048.
%K A320251 nonn,tabl
%O A320251 0,6
%A A320251 _Ilya Gutkovskiy_, Oct 08 2018