cp's OEIS Frontend

A288456 Chebyshev coefficients of density of states of honeycomb lattice.

%I A288456 #12 Jun 19 2017 15:15:05
%S A288456 1,-3,-15,141,-1503,9117,-46959,2349,1947969,-26479299,166125105,
%T A288456 -476958771,-7411008159,122517898461,-1220344831791,7016585864301,
%U A288456 -14334148360575,-334610402172291,4919241139007601,-42532841711020275,172482611175249057,717799148664446493,-24646866746992333551
%N A288456 Chebyshev coefficients of density of states of honeycomb lattice.
%C A288456 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 honeycomb lattice (z=3), 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 A288456 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 A288456 The author was unable to obtain a closed form for z^n g_n.
%H A288456 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 A288456 Whon[n_] := If[OddQ[n], 0,
%t A288456    Sum[Binomial[n/2, j]^2 Binomial[2 j, j], {j, 0, n/2}]];
%t A288456 ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
%t A288456 zng[n_] := Sum[ank[n, k]*3^(n - k)*Whon[k], {k, 0, n}];
%t A288456 Table[zng[n], {n,0,50}]
%Y A288456 Related to numbers of walks returning to origin, W_n, on honeycomb lattice (A002893).
%Y A288456 See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
%K A288456 sign
%O A288456 0,2
%A A288456 _Yen-Lee Loh_, Jun 16 2017