OFFSET
0,3
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 triangular lattice, 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.
MATHEMATICA
Whon[n_] := If[OddQ[n], 0, Sum[Binomial[n/2, j]^2 Binomial[2j, j], {j, 0, n/2}]];
Wtri[n_] := Sum[Binomial[n, j] (-3)^(n-j) Whon[2j], {j, 0, n}];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*6^(n - k)*Wtri[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]
CROSSREFS
Related to numbers of walks returning to origin, W_n, on triangular lattice (A002898).
KEYWORD
sign
AUTHOR
Yen-Lee Loh, Jun 16 2017
STATUS
approved