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%I #16 Jun 19 2017 15:16:08
%S 1,-48,1728,-79872,4058112,-216956928,11977752576,-676117610496,
%T 38792847949824,-2253773963526144,132241430641901568,
%U -7821943674035503104,465750331610495975424,-27888626411947306254336,1677958399935741979262976,-101375476324084742212288512,6146869366762959307806867456
%N Chebyshev coefficients of density of states of BCC lattice.
%C 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 body-centered cubic lattice (z=8), 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 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 For the bcc lattice (z=8), 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)^3 * 4F3 (-N, -N, -N, -N; 1-2N, 1/2-N, 1/2-N; 1).
%H 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 zng[n_] :=
%t If[OddQ[n], 0,
%t (1 + KroneckerDelta[m]) 2^(2 m - 1) *
%t Binomial[2m,m]^3 HypergeometricPFQ[{-m,-m,-m,-m}, {1-2m, 1/2-m, 1/2-m}, 1] /. m -> n/2];
%t Table[zng[n], {n,0,50}]
%t Wchain[n_] := If[OddQ[n], 0, Binomial[n, n/2]];
%t Wbcc[n_] := Wchain[n]^3;
%t ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
%t zng[n_] := Sum[ank[n, k]*8^(n-k)*Wbcc[k], {k, 0, n}];
%t Table[zng[n], {n,0,50}]
%Y Related to numbers of walks returning to origin, W_n, on BCC lattice (A002897).
%Y See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
%K sign
%O 0,2
%A _Yen-Lee Loh_, Jun 16 2017
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