

A288454


Chebyshev coefficients of density of states of square lattice.


8



1, 8, 32, 512, 4608, 73728, 819200, 13107200, 160563200, 2569011200, 33294385152, 532710162432, 7161992183808, 114591874940928, 1580900152246272, 25294402435940352, 355702534255411200, 5691240548086579200, 81223136710964019200, 1299570187375424307200, 18765793505701126995968
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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 square 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.
For the square lattice (z=4), the even Chebyshev coefficients can be expressed in closed form in terms of the hypergeometric function pFq, as z^{2N} g_{2N} = (1 + delta_N) * 2^(2N1) Binomial(2N,N)^2 * 3F2 (N, N, N; 12N, 1/2N; 1).


LINKS

Table of n, a(n) for n=0..20.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [mathph], 2017.


MATHEMATICA

zng[n_] := If[OddQ[n], 0, (1+KroneckerDelta[m]) 2^(2m1) Binomial[2m, m]^2 HypergeometricPFQ[{m, m, m}, {12m, 1/2m}, 1] /. m>n/2];
Table[zng[n], {n, 0, 50}]
Wchain[n_] := If[OddQ[n], 0, Binomial[n, n/2]];
Wsq[n_] := Wchain[n]^2;
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*4^(n  k)*Wsq[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]


CROSSREFS

Related to numbers of walks returning to origin, W_n, on square lattice (A002894).
See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
Sequence in context: A214539 A140789 A120781 * A139286 A139306 A214594
Adjacent sequences: A288451 A288452 A288453 * A288455 A288456 A288457


KEYWORD

sign


AUTHOR

YenLee Loh, Jun 16 2017


STATUS

approved



