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.

1, 1, 1, 1, 4, 7, 1, 9, 39, 87, 1, 16, 126, 608, 1553, 1, 25, 310, 2470, 12985, 36145, 1, 36, 645, 7560, 62595, 351252, 1037367, 1, 49, 1197, 19285, 225715, 1946259, 11481631, 35402983, 1, 64, 2044, 43232, 673190, 8011136, 71657404, 439552864, 1400424097

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 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).

Examples

			Triangle starts:
[0] 1;
[1] 1,  1;
[2] 1,  4,    7;
[3] 1,  9,   39,    87;
[4] 1, 16,  126,   608,   1553;
[5] 1, 25,  310,  2470,  12985,   36145;
[6] 1, 36,  645,  7560,  62595,  351252,  1037367;
[7] 1, 49, 1197, 19285, 225715, 1946259, 11481631,  35402983;
[8] 1, 64, 2044, 43232, 673190, 8011136, 71657404, 439552864, 1400424097;
		

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), this sequence (type B), A343806 (type C), A343807 (type D).

Programs

Maple

alias(W = LambertW):
EhrB := exp(-W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):
ser := series(EhrB, 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..9);

Mathematica

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