A288460 Chebyshev coefficients of density of states of triangular lattice.

1, 0, -24, 48, 288, -2880, 3072, 64512, -400896, -245760, 12496896, -50688000, -159547392, 2133540864, -4964253696, -42047373312, 313193005056, -179042254848, -8158768005120, 36487616790528, 65397155954688, -1204277276049408, 2427936640598016, 19127143199932416, -107713462133587968, -223101299070074880

Offset: 0

Author

Yen-Lee Loh, Jun 16 2017

Keywords

• sign

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 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.

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

• Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.

Crossrefs

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

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

Programs

Mathematica

Whon[n_] := If[OddQ[n], 0, Sum[Binomial[n/2,j]^2 Binomial[2j,j], {j, 0, n/2}]];
Wtri[n_] := Sum[Binomial[n,j] (-3)^(n-j) Whon[2j], {j, 0, n}];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*6^(n - k)*Wtri[k], {k, 0, n}];
Table[zng[n], {n,0,50}]