

A288461


Chebyshev coefficients of density of states of FCC lattice.


8



1, 0, 120, 192, 11232, 69120, 887808, 11870208, 34721280, 1458585600, 4612792320, 124992552960, 1294622392320, 3918630223872, 170411025235968, 1023380614545408, 11537631123996672, 248923798395420672, 709263007722504192, 30740965743050883072, 367936297318798589952, 1683415347595793399808
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

This is the sequence of integers z^n g_n for n=0,1,2,3,... where g_n are the coefficients in the Chebyshev polynomial expansion of the density of states of the facecentered cubic lattice (z=12), g(w) = 1 / (Pi*sqrt(1w^2)) * Sum_{n>=0} (2delta_n) g_n T_n(w). Here w <= 1 and delta is the Kronecker delta. The coordination number is z=6. Note that the triangular lattice is sometimes called the hexagonal lattice.
The Chebyshev coefficients, g_n, are related to the number of walks on the lattice that return to the origin, W_n, as g_n = Sum_{k=0..n} a_{nk} z^{k} W_k, where z is the coordination number of the lattice and a_{nk} are the coefficients of Chebyshev polynomials such that T_n(x) = Sum_{k=0..n} a_{nk} x^k. The author was unable to obtain a closed form for z^n g_n.


LINKS

Table of n, a(n) for n=0..21.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [mathph], 2017.


MATHEMATICA

Wdia[n_] := If[OddQ[n], 0,
Sum[Binomial[n/2, j]^2 Binomial[2j, j] Binomial[n2j, n/2j], {j, 0, n/2}]];
Wfcc[n_] := Sum[Binomial[n, j] (4)^(nj) Wdia[2 j], {j, 0, n}];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*12^(nk)*Wfcc[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]


CROSSREFS

Related to numbers of walks returning to origin, W_n, on FCC lattice (A002899).
Cf. A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
Sequence in context: A247851 A179232 A056994 * A114823 A069790 A064224
Adjacent sequences: A288458 A288459 A288460 * A288462 A288463 A288464


KEYWORD

sign


AUTHOR

YenLee Loh, Jun 19 2017


STATUS

approved



