|
%I #11 Jun 19 2017 18:27:31
%S 1,0,-120,192,11232,-69120,-887808,11870208,34721280,-1458585600,
%T 4612792320,124992552960,-1294622392320,-3918630223872,
%U 170411025235968,-1023380614545408,-11537631123996672,248923798395420672,-709263007722504192,-30740965743050883072,367936297318798589952,1683415347595793399808
%N Chebyshev coefficients of density of states of FCC lattice.
%C 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 face-centered cubic lattice (z=12), g(w) = 1 / (Pi*sqrt(1-w^2)) * Sum_{n>=0} (2-delta_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.
%C 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.
%H Yen Lee Loh, <a href="http://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.
%t Wdia[n_] := If[OddQ[n], 0,
%t Sum[Binomial[n/2,j]^2 Binomial[2j,j] Binomial[n-2j, n/2-j], {j, 0, n/2}]];
%t Wfcc[n_] := Sum[Binomial[n, j] (-4)^(n-j) Wdia[2 j], {j, 0, n}];
%t ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
%t zng[n_] := Sum[ank[n, k]*12^(n-k)*Wfcc[k], {k, 0, n}];
%t Table[zng[n], {n,0,50}]
%Y Related to numbers of walks returning to origin, W_n, on FCC lattice (A002899).
%Y Cf. A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
%K sign
%O 0,3
%A _Yen-Lee Loh_, Jun 19 2017
|
