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.

1, 1, 2, 1, 6, 14, 1, 12, 66, 172, 1, 20, 192, 1080, 3036, 1, 30, 440, 4040, 23580, 69976, 1, 42, 870, 11600, 106620, 644568, 1991656, 1, 56, 1554, 28140, 364140, 3396960, 21170520, 67484880, 1, 72, 2576, 60592, 1037400, 13362272, 126973504, 811924032, 2652878864

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 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).

Examples

			Triangle starts:
[0] 1;
[1] 1,  2;
[2] 1,  6,   14;
[3] 1, 12,   66,   172;
[4] 1, 20,  192,  1080,    3036;
[5] 1, 30,  440,  4040,   23580,    69976;
[6] 1, 42,  870, 11600,  106620,   644568,   1991656;
[7] 1, 56, 1554, 28140,  364140,  3396960,  21170520,  67484880;
[8] 1, 72, 2576, 60592, 1037400, 13362272, 126973504, 811924032, 2652878864;
		

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), this sequence (type C), A343807 (type D).

Programs

Maple

alias(W = LambertW):
EhrC := exp(-(t+1)*W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):
ser := series(EhrC, 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[CoefficientList[cx[n], t], {n, 0, 8}] // Flatten