A288460 - OEIS

%I #12 Jun 19 2017 15:15:18

%S 1,0,-24,48,288,-2880,3072,64512,-400896,-245760,12496896,-50688000,

%T -159547392,2133540864,-4964253696,-42047373312,313193005056,

%U -179042254848,-8158768005120,36487616790528,65397155954688,-1204277276049408,2427936640598016,19127143199932416,-107713462133587968,-223101299070074880

%N Chebyshev coefficients of density of states of triangular 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 triangular lattice, 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.

%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 Whon[n_] := If[OddQ[n], 0, Sum[Binomial[n/2,j]^2 Binomial[2j,j], {j, 0, n/2}]];

%t Wtri[n_] := Sum[Binomial[n,j] (-3)^(n-j) Whon[2j], {j, 0, n}];

%t ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];

%t zng[n_] := Sum[ank[n, k]*6^(n - k)*Wtri[k], {k, 0, n}];

%t Table[zng[n], {n,0,50}]

%Y Related to numbers of walks returning to origin, W_n, on triangular lattice (A002898).

%Y See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.

%K sign

%O 0,3

%A _Yen-Lee Loh_, Jun 16 2017