A288461 Chebyshev coefficients of density of states of FCC lattice.

1, 0, -120, 192, 11232, -69120, -887808, 11870208, 34721280, -1458585600, 4612792320, 124992552960, -1294622392320, -3918630223872, 170411025235968, -1023380614545408, -11537631123996672, 248923798395420672, -709263007722504192, -30740965743050883072, 367936297318798589952, 1683415347595793399808

Offset: 0

Author

Yen-Lee Loh, Jun 19 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 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.

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 FCC lattice (A002899).

Cf. A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.

Programs

Mathematica

Wdia[n_] := If[OddQ[n], 0,
   Sum[Binomial[n/2,j]^2 Binomial[2j,j] Binomial[n-2j, n/2-j], {j, 0, n/2}]];
Wfcc[n_] := Sum[Binomial[n, j] (-4)^(n-j) Wdia[2 j], {j, 0, n}];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*12^(n-k)*Wfcc[k], {k, 0, n}];
Table[zng[n], {n,0,50}]