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A343807 T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 - 2*t + V))/(4*t))*sqrt(1 + V)) where V = W(-2*t*x) and W denotes the Lambert function. Table read by rows, T(n, k) for 0 <= k <= n.

%I A343807 #12 May 02 2021 06:29:58
%S A343807 1,1,0,1,2,2,1,6,18,32,1,12,72,280,636,1,20,200,1320,6060,15744,1,30,
%T A343807 450,4480,32460,166536,470680,1,42,882,12320,127260,996408,5526136,
%U A343807 16542336,1,56,1568,29232,405720,4384800,36529920,214436160,669165840
%N A343807 T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 - 2*t + V))/(4*t))*sqrt(1 + V)) where V = W(-2*t*x) and W denotes the Lambert function. Table read by rows, T(n, k) for 0 <= k <= n.
%C A343807 The rows of the triangle give the coefficients of the Ehrhart polynomials of integral Coxeter permutahedra of type D. These polynomials count lattice points in a dilated lattice polytope. For a definition see Ardila et al. (p. 1158), the generating functions of these polynomials for the classical root systems are given in theorem 5.2 (p. 1163).
%H A343807 Federico Ardila, Matthias Beck, and Jodi McWhirter, <a href="https://doi.org/10.18257/raccefyn.1189">The arithmetic of Coxeter permutahedra</a>, Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 44(173):1152-1166, 2020.
%e A343807 [0] 1;
%e A343807 [1] 1,  0;
%e A343807 [2] 1,  2,    2;
%e A343807 [3] 1,  6,   18,    32;
%e A343807 [4] 1, 12,   72,   280,    636;
%e A343807 [5] 1, 20,  200,  1320,   6060,   15744;
%e A343807 [6] 1, 30,  450,  4480,  32460,  166536,   470680;
%e A343807 [7] 1, 42,  882, 12320, 127260,  996408,  5526136,  16542336;
%e A343807 [8] 1, 56, 1568, 29232, 405720, 4384800, 36529920, 214436160, 669165840;
%p A343807 alias(W = LambertW):
%p A343807 EhrD := exp(-(1-t)*W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t)) / sqrt(1+W(-2*t*x)):
%p A343807 ser := series(EhrD, x, 10): cx := n -> n!*coeff(ser, x, n):
%p A343807 T := n -> seq(coeff(cx(n), t, k), k = 0..n): seq(T(n), n = 0..8);
%t A343807 P := ProductLog[-2 t x]; gf := 1/(E^((P (2 - 2 t + P))/(4 t)) Sqrt[1 + P]);
%t A343807 ser := Series[gf, {x, 0, 10}];  cx[n_] := n! Coefficient[ser, x, n];
%t A343807 Table[If[n == 1, {1, 0}, CoefficientList[cx[n], t]], {n, 0, 8}] // Flatten
%Y A343807 Cf. A138464 (type A), A343805 (type B), A343806 (type C), this sequence (type D).
%K A343807 nonn,tabl
%O A343807 0,5
%A A343807 _Peter Luschny_, May 01 2021