

A288456


Chebyshev coefficients of density of states of honeycomb lattice.


8



1, 3, 15, 141, 1503, 9117, 46959, 2349, 1947969, 26479299, 166125105, 476958771, 7411008159, 122517898461, 1220344831791, 7016585864301, 14334148360575, 334610402172291, 4919241139007601, 42532841711020275, 172482611175249057, 717799148664446493, 24646866746992333551
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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(1w^2)) * Sum_{n>=0} (2delta_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 [mathph], 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

YenLee Loh, Jun 16 2017


STATUS

approved



