login
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.
3
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
0,3
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).
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.
EXAMPLE
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;
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
CROSSREFS
Cf. A138464 (type A), A343805 (type B), this sequence (type C), A343807 (type D).
Sequence in context: A136456 A123968 A282329 * A372254 A210654 A068797
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 01 2021
STATUS
approved