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A343806 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 A343806 #12 May 02 2021 06:30:06
%S A343806 1,1,2,1,6,14,1,12,66,172,1,20,192,1080,3036,1,30,440,4040,23580,
%T A343806 69976,1,42,870,11600,106620,644568,1991656,1,56,1554,28140,364140,
%U A343806 3396960,21170520,67484880,1,72,2576,60592,1037400,13362272,126973504,811924032,2652878864
%N A343806 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 A343806 The rows of the triangle give the coefficients of the Ehrhart polynomials of integral Coxeter permutahedra of type C. 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 A343806 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 A343806 Triangle starts:
%e A343806 [0] 1;
%e A343806 [1] 1,  2;
%e A343806 [2] 1,  6,   14;
%e A343806 [3] 1, 12,   66,   172;
%e A343806 [4] 1, 20,  192,  1080,    3036;
%e A343806 [5] 1, 30,  440,  4040,   23580,    69976;
%e A343806 [6] 1, 42,  870, 11600,  106620,   644568,   1991656;
%e A343806 [7] 1, 56, 1554, 28140,  364140,  3396960,  21170520,  67484880;
%e A343806 [8] 1, 72, 2576, 60592, 1037400, 13362272, 126973504, 811924032, 2652878864;
%p A343806 alias(W = LambertW):
%p A343806 EhrC := exp(-(t+1)*W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):
%p A343806 ser := series(EhrC, x, 10): cx := n -> n!*coeff(ser, x, n):
%p A343806 T := n -> seq(coeff(cx(n), t, k), k=0..n): seq(T(n), n = 0..8);
%t A343806 P := ProductLog[-2 t x]; gf := 1/(E^((P (2 + 2 t + P))/(4 t)) Sqrt[1 + P]);
%t A343806 ser := Series[gf, {x, 0, 10}]; cx[n_] := n! Coefficient[ser, x, n];
%t A343806 Table[CoefficientList[cx[n], t], {n, 0, 8}] // Flatten
%Y A343806 Cf. A138464 (type A), A343805 (type B), this sequence (type C), A343807 (type D).
%K A343806 nonn,tabl
%O A343806 0,3
%A A343806 _Peter Luschny_, May 01 2021