cp's OEIS Frontend

A288454 Chebyshev coefficients of density of states of square lattice.

%I A288454 #18 Jun 19 2017 15:16:58
%S A288454 1,-8,32,-512,4608,-73728,819200,-13107200,160563200,-2569011200,
%T A288454 33294385152,-532710162432,7161992183808,-114591874940928,
%U A288454 1580900152246272,-25294402435940352,355702534255411200,-5691240548086579200,81223136710964019200,-1299570187375424307200,18765793505701126995968
%N A288454 Chebyshev coefficients of density of states of square lattice.
%C A288454 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(1-w^2)) * Sum_{n>=0} (2-delta_n) g_n T_n(w). Here |w| <= 1 and delta is the Kronecker delta.
%C A288454 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.
%C A288454 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^(2N-1) Binomial(2N,N)^2 * 3F2 (-N, -N, -N; 1-2N, 1/2-N; 1).
%H A288454 Yen Lee Loh, <a href="http://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.
%t A288454 zng[n_] := If[OddQ[n], 0, (1+KroneckerDelta[m]) 2^(2m-1) Binomial[2m, m]^2 HypergeometricPFQ[{-m, -m, -m}, {1-2m,1/2-m}, 1] /. m->n/2];
%t A288454 Table[zng[n], {n,0,50}]
%t A288454 Wchain[n_] := If[OddQ[n], 0, Binomial[n, n/2]];
%t A288454 Wsq[n_] := Wchain[n]^2;
%t A288454 ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
%t A288454 zng[n_] := Sum[ank[n, k]*4^(n - k)*Wsq[k], {k, 0, n}];
%t A288454 Table[zng[n], {n,0,50}]
%Y A288454 Related to numbers of walks returning to origin, W_n, on square lattice (A002894).
%Y A288454 See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
%K A288454 sign
%O A288454 0,2
%A A288454 _Yen-Lee Loh_, Jun 16 2017