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

0,5

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).

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

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;

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

Crossrefs

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

Sequence in context: A371333 A019670 A093436 * A082169 A209634 A340584

Adjacent sequences: A343802 A343803 A343804 * A343806 A343807 A343808

Keyword

nonn,tabl

Author

Peter Luschny, May 01 2021

Status

approved