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A288455
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Chebyshev coefficients of density of states of BCC lattice.
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8
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1, -48, 1728, -79872, 4058112, -216956928, 11977752576, -676117610496, 38792847949824, -2253773963526144, 132241430641901568, -7821943674035503104, 465750331610495975424, -27888626411947306254336, 1677958399935741979262976, -101375476324084742212288512, 6146869366762959307806867456
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OFFSET
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0,2
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COMMENTS
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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.
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 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).
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LINKS
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MATHEMATICA
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zng[n_] :=
If[OddQ[n], 0,
(1 + KroneckerDelta[m]) 2^(2 m - 1) *
Binomial[2m, m]^3 HypergeometricPFQ[{-m, -m, -m, -m}, {1-2m, 1/2-m, 1/2-m}, 1] /. m -> n/2];
Table[zng[n], {n, 0, 50}]
Wchain[n_] := If[OddQ[n], 0, Binomial[n, n/2]];
Wbcc[n_] := Wchain[n]^3;
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*8^(n-k)*Wbcc[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 BCC lattice (A002897).
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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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