A343806 - OEIS

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

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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

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, 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

Adjacent sequences: A343803 A343804 A343805 * A343807 A343808 A343809

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 01 2021

STATUS

approved