A288456 Chebyshev coefficients of density of states of honeycomb lattice.

1, -3, -15, 141, -1503, 9117, -46959, 2349, 1947969, -26479299, 166125105, -476958771, -7411008159, 122517898461, -1220344831791, 7016585864301, -14334148360575, -334610402172291, 4919241139007601, -42532841711020275, 172482611175249057, 717799148664446493, -24646866746992333551

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 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.

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..22.

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

Mathematica

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

Crossrefs

Related to numbers of walks returning to origin, W_n, on honeycomb lattice (A002893).

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

Sequence in context: A179471 A203417 A086228 * A230367 A293844 A181535

Adjacent sequences: A288453 A288454 A288455 * A288457 A288458 A288459

Keyword

sign

Author

Yen-Lee Loh, Jun 16 2017

Status

approved