

A288457


Chebyshev coefficients of density of states of diamond lattice.


8



1, 8, 32, 1024, 12800, 90112, 131072, 2097152, 78774272, 3080716288, 49736056832, 407753457664, 222801428480, 19645180411904, 494299196162048, 22797274090307584, 393216908922454016, 3294704322255781888, 1334801068806111232, 228652837223366918144, 4282607861714030428160, 222230748909257887842304
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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 diamond lattice (z=4), 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..21.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [mathph], 2017.


MATHEMATICA

Wdia[n_] := If[OddQ[n], 0,
Sum[Binomial[n/2, j]^2 Binomial[2j, j] Binomial[n2j, n/2j], {j, 0, n/2}]];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*4^(nk)*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

YenLee Loh, Jun 16 2017


STATUS

approved



