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

Author

Stefano Spezia, Jun 17 2021

Keywords

• nonn
• tabl

Examples

			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 ...
...
		

Links

• 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.

Crossrefs

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

Programs

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}]]