|
%I #12 Jun 19 2017 15:15:47
%S 1,-8,-32,1024,-12800,90112,-131072,-2097152,-78774272,3080716288,
%T -49736056832,407753457664,-222801428480,-19645180411904,
%U -494299196162048,22797274090307584,-393216908922454016,3294704322255781888,1334801068806111232,-228652837223366918144,-4282607861714030428160,222230748909257887842304
%N Chebyshev coefficients of density of states of diamond lattice.
%C This is the sequence of integers z^n g_n for n=0,2,4,6,... where g_n are the coefficients in the Chebyshev polynomial expansion of the density of states of the diamond lattice (z=4), 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.
%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.
%C 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 ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
%t zng[n_] := Sum[ank[n, k]*4^(n-k)*Wdia[k], {k, 0, n}];
%t Table[zng[n], {n,0,50}]
%Y Related to numbers of walks returning to origin, W_n, on diamond lattice (A002895).
%Y See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
%K sign
%O 0,2
%A _Yen-Lee Loh_, Jun 16 2017
|
