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%I #10 Jun 30 2021 19:56:29
%S 0,1,0,4,1,0,13,12,1,0,40,109,28,1,0,121,888,493,60,1,0,364,6841,7192,
%T 1837,124,1,0,1093,51012,95161,42840,6253,252,1,0,3280,372709,1189108,
%U 865081,220120,20269,508,1,0,9841,2687088,14331493,16022100,6396601,1040088,63853,1020,1,0
%N 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.
%H Beáta Bényi and Toshiki Matsusaka, <a href="https://arxiv.org/abs/2106.05585">Extensions of the combinatorics of poly-Bernoulli numbers</a>, arXiv:2106.05585 [math.CO], 2021. See p. 9.
%H Takao Komatsu, <a href="https://arxiv.org/abs/1806.05515">On poly-Euler numbers of the second kind</a>, arXiv:1806.05515 [math.NT], 2018.
%e n\k| 0 1 2 3 4 ...
%e ---+------------------------------
%e 0 | 0 0 0 0 0 ...
%e 1 | 1 1 1 1 1 ...
%e 2 | 4 12 28 60 124 ...
%e 3 | 13 109 493 1837 6253 ...
%e 4 | 40 888 7192 42840 220120 ...
%e ...
%t 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}]]
%Y Cf. A000004 (n = 0), A000012 (n = 1), A003462 (k = 0), A081200 (k = 1), A345394.
%K nonn,tabl
%O 0,4
%A _Stefano Spezia_, Jun 17 2021
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