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A343805 T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 + 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 A343805 #14 May 02 2021 06:29:27
%S A343805 1,1,1,1,4,7,1,9,39,87,1,16,126,608,1553,1,25,310,2470,12985,36145,1,
%T A343805 36,645,7560,62595,351252,1037367,1,49,1197,19285,225715,1946259,
%U A343805 11481631,35402983,1,64,2044,43232,673190,8011136,71657404,439552864,1400424097
%N A343805 T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 + 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 A343805 The rows of the triangle give the coefficients of the Ehrhart polynomials of integral Coxeter permutahedra of type B. 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 A343805 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 A343805 Triangle starts:
%e A343805 [0] 1;
%e A343805 [1] 1,  1;
%e A343805 [2] 1,  4,    7;
%e A343805 [3] 1,  9,   39,    87;
%e A343805 [4] 1, 16,  126,   608,   1553;
%e A343805 [5] 1, 25,  310,  2470,  12985,   36145;
%e A343805 [6] 1, 36,  645,  7560,  62595,  351252,  1037367;
%e A343805 [7] 1, 49, 1197, 19285, 225715, 1946259, 11481631,  35402983;
%e A343805 [8] 1, 64, 2044, 43232, 673190, 8011136, 71657404, 439552864, 1400424097;
%p A343805 alias(W = LambertW):
%p A343805 EhrB := exp(-W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):
%p A343805 ser := series(EhrB, x, 10): cx := n -> n!*coeff(ser, x, n):
%p A343805 T := n -> seq(coeff(cx(n), t, k), k=0..n): seq(T(n), n = 0..9);
%t A343805 P := ProductLog[-2 t x]; gf := 1/(E^((P (2 + P))/(4 t)) Sqrt[1 + P]);
%t A343805 ser := Series[gf, {x, 0, 10}]; cx[n_] := n! Coefficient[ser, x, n];
%t A343805 Table[CoefficientList[cx[n], t], {n, 0, 8}] // Flatten
%Y A343805 Cf. A138464 (type A), this sequence (type B), A343806 (type C), A343807 (type D).
%K A343805 nonn,tabl
%O A343805 0,5
%A A343805 _Peter Luschny_, May 01 2021