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

0,5

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).

Links

Table of n, a(n) for n=0..44.

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

Example

[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;

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

Crossrefs

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

Sequence in context: A350297 A181731 A278792 * A340734 A108338 A021455

Adjacent sequences: A343804 A343805 A343806 * A343808 A343809 A343810

Keyword

nonn,tabl

Author

Peter Luschny, May 01 2021

Status

approved