A345394 Array read by ascending antidiagonals: A(n, k) = n!*[x^n] Li(-k, 1 - exp(-4*x))/(4*sinh(x)), where Li(n, z) is the polylogarithm function.

1, 2, 1, 5, 6, 1, 14, 37, 14, 1, 41, 234, 165, 30, 1, 122, 1513, 1826, 613, 62, 1, 365, 9966, 19689, 10770, 2085, 126, 1, 1094, 66637, 210134, 175465, 55154, 6757, 254, 1, 3281, 450834, 2236365, 2741670, 1287657, 260274, 21285, 510, 1, 9842, 3077713, 23819306, 41809933, 27930182, 8420713, 1167026, 65893, 1022, 1

Offset

0,2

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.

Komatsu, Takao Complementary Euler numbers. Period. Math. Hung. 75, No. 2, 302-314 (2017).

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  |   1     1      1       1        1 ...
1  |   2     6     14      30       62 ...
2  |   5    37    165     613     2085 ...
3  |  14   234   1826   10770    55154 ...
4  |  41  1513  19689  175465  1287657 ...
...

Mathematica

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

Crossrefs

Cf. A000012 (n = 0), A007051 (k = 0), A081188 (k = 1), A305861 (n = 2), A305862 (n = 3), A305863 (n = 4), A316526 (n = 5), A345393.

Sequence in context: A120986 A095801 A128567 * A217204 A179455 A039810

Adjacent sequences: A345391 A345392 A345393 * A345395 A345396 A345397

Keyword

nonn,tabl

Author

Stefano Spezia, Jun 17 2021

Status

approved