A288457 Chebyshev coefficients of density of states of diamond lattice.

1, -8, -32, 1024, -12800, 90112, -131072, -2097152, -78774272, 3080716288, -49736056832, 407753457664, -222801428480, -19645180411904, -494299196162048, 22797274090307584, -393216908922454016, 3294704322255781888, 1334801068806111232, -228652837223366918144, -4282607861714030428160, 222230748909257887842304

Offset

0,2

Comments

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.

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

Table of n, a(n) for n=0..21.

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

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}]];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*4^(n-k)*Wdia[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]

Crossrefs

Related to numbers of walks returning to origin, W_n, on diamond lattice (A002895).

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

Sequence in context: A139286 A139306 A214594 * A166995 A079271 A336220

Adjacent sequences: A288454 A288455 A288456 * A288458 A288459 A288460

Keyword

sign

Author

Yen-Lee Loh, Jun 16 2017

Status

approved