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.

1, 1, 0, 1, 2, 2, 1, 6, 18, 32, 1, 12, 72, 280, 636, 1, 20, 200, 1320, 6060, 15744, 1, 30, 450, 4480, 32460, 166536, 470680, 1, 42, 882, 12320, 127260, 996408, 5526136, 16542336, 1, 56, 1568, 29232, 405720, 4384800, 36529920, 214436160, 669165840

Offset: 0

Author

Peter Luschny, May 01 2021

Keywords

• nonn
• tabl

Comments

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).

Examples

			[0] 1;
[1] 1,  0;
[2] 1,  2,    2;
[3] 1,  6,   18,    32;
[4] 1, 12,   72,   280,    636;
[5] 1, 20,  200,  1320,   6060,   15744;
[6] 1, 30,  450,  4480,  32460,  166536,   470680;
[7] 1, 42,  882, 12320, 127260,  996408,  5526136,  16542336;
[8] 1, 56, 1568, 29232, 405720, 4384800, 36529920, 214436160, 669165840;
		

Links

• Federico Ardila, Matthias Beck, and Jodi McWhirter, The arithmetic of Coxeter permutahedra, Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 44(173):1152-1166, 2020.

Crossrefs

Cf. A138464 (type A), A343805 (type B), A343806 (type C), this sequence (type D).

Programs

Maple

alias(W = LambertW):
EhrD := exp(-(1-t)*W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t)) / sqrt(1+W(-2*t*x)):
ser := series(EhrD, x, 10): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k = 0..n): seq(T(n), n = 0..8);

Mathematica

P := ProductLog[-2 t x]; gf := 1/(E^((P (2 - 2 t + P))/(4 t)) Sqrt[1 + P]);
ser := Series[gf, {x, 0, 10}];  cx[n_] := n! Coefficient[ser, x, n];
Table[If[n == 1, {1, 0}, CoefficientList[cx[n], t]], {n, 0, 8}] // Flatten