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

Author

Stefano Spezia, Jun 17 2021

Keywords

• nonn
• tabl

Examples

			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 ...
...
		

Links

• 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.

Crossrefs

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

Programs

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}]]