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A288461
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Chebyshev coefficients of density of states of FCC lattice.
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8
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1, 0, -120, 192, 11232, -69120, -887808, 11870208, 34721280, -1458585600, 4612792320, 124992552960, -1294622392320, -3918630223872, 170411025235968, -1023380614545408, -11537631123996672, 248923798395420672, -709263007722504192, -30740965743050883072, 367936297318798589952, 1683415347595793399808
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OFFSET
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0,3
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COMMENTS
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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 face-centered cubic lattice (z=12), 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.
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LINKS
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MATHEMATICA
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Wdia[n_] := If[OddQ[n], 0,
Sum[Binomial[n/2, j]^2 Binomial[2j, j] Binomial[n-2j, n/2-j], {j, 0, n/2}]];
Wfcc[n_] := Sum[Binomial[n, j] (-4)^(n-j) Wdia[2 j], {j, 0, n}];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*12^(n-k)*Wfcc[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]
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CROSSREFS
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Related to numbers of walks returning to origin, W_n, on FCC lattice (A002899).
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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