A345393 Array read by ascending antidiagonals: A(n, k) = n!*[x^(n-1)] Li(-k, 1 - exp(-4*x))/(4*x*cosh(x)), where Li(n, z) is the polylogarithm function.

0, 1, 0, 4, 1, 0, 13, 12, 1, 0, 40, 109, 28, 1, 0, 121, 888, 493, 60, 1, 0, 364, 6841, 7192, 1837, 124, 1, 0, 1093, 51012, 95161, 42840, 6253, 252, 1, 0, 3280, 372709, 1189108, 865081, 220120, 20269, 508, 1, 0, 9841, 2687088, 14331493, 16022100, 6396601, 1040088, 63853, 1020, 1, 0

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021. See p. 9.

Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018.

Example

n\k|   0    1     2      3       4 ...
---+------------------------------
0  |   0    0     0      0       0 ...
1  |   1    1     1      1       1 ...
2  |   4   12    28     60     124 ...
3  |  13  109   493   1837    6253 ...
4  |  40  888  7192  42840  220120 ...
...

Mathematica

A[n_, k_]:=n!Coefficient[Series[PolyLog[-k, 1-Exp[-4x]]/(4x Cosh[x]), {x, 0, n}], x, n-1]; Flatten[Table[A[n-k, k], {n, 0, 9}, {k, 0, n}]]

Crossrefs

Cf. A000004 (n = 0), A000012 (n = 1), A003462 (k = 0), A081200 (k = 1), A345394.

Sequence in context: A060638 A244125 A007789 * A081114 A069018 A156811

Adjacent sequences: A345390 A345391 A345392 * A345394 A345395 A345396

Keyword

nonn,tabl

Author

Stefano Spezia, Jun 17 2021

Status

approved