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).

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, 33, 114, 207, 221, 144, 64, 1, 1, 65, 308, 723, 991, 776, 377, 128, 1, 1, 129, 858, 2631, 4805, 4752, 2725, 987, 256, 1, 1, 257, 2444, 9843, 24655, 31880, 22769, 9569, 2584, 512

Offset

0,6

Comments

A(n,k) is the invert transform of k-th powers evaluated at n.

Links

Table of n, a(n) for n=0..65.

N. J. A. Sloane, Transforms

Formula

G.f. of column k: 1/(1 - PolyLog(-k,x)), where PolyLog() is the polylogarithm function.

Example

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 + ...
Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   8,   18,   44,   114,    308,  ...
   8,  21,   63,  207,   723,   2631,  ...
  16,  55,  221,  991,  4805,  24655,  ...

Mathematica

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
Table[Function[k, SeriesCoefficient[1/(1 - PolyLog[-k, x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..3 give A011782, A088305, A033453, A144109.

Main diagonal gives A301655.

Cf. A144048.

Sequence in context: A336187 A171881 A321877 * A210341 A160449 A326322

Adjacent sequences: A320248 A320249 A320250 * A320252 A320253 A320254

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Oct 08 2018

Status

approved