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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.
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%I #14 May 02 2021 06:29:27

%S 1,1,1,1,4,7,1,9,39,87,1,16,126,608,1553,1,25,310,2470,12985,36145,1,

%T 36,645,7560,62595,351252,1037367,1,49,1197,19285,225715,1946259,

%U 11481631,35402983,1,64,2044,43232,673190,8011136,71657404,439552864,1400424097

%N 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 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 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 Triangle starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 4, 7;

%e [3] 1, 9, 39, 87;

%e [4] 1, 16, 126, 608, 1553;

%e [5] 1, 25, 310, 2470, 12985, 36145;

%e [6] 1, 36, 645, 7560, 62595, 351252, 1037367;

%e [7] 1, 49, 1197, 19285, 225715, 1946259, 11481631, 35402983;

%e [8] 1, 64, 2044, 43232, 673190, 8011136, 71657404, 439552864, 1400424097;

%p alias(W = LambertW):

%p EhrB := exp(-W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):

%p ser := series(EhrB, x, 10): cx := n -> n!*coeff(ser, x, n):

%p T := n -> seq(coeff(cx(n), t, k), k=0..n): seq(T(n), n = 0..9);

%t P := ProductLog[-2 t x]; gf := 1/(E^((P (2 + P))/(4 t)) Sqrt[1 + P]);

%t ser := Series[gf, {x, 0, 10}]; cx[n_] := n! Coefficient[ser, x, n];

%t Table[CoefficientList[cx[n], t], {n, 0, 8}] // Flatten

%Y Cf. A138464 (type A), this sequence (type B), A343806 (type C), A343807 (type D).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, May 01 2021