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A288454
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Chebyshev coefficients of density of states of square lattice.
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8
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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
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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 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.
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^(2N-1) Binomial(2N,N)^2 * 3F2 (-N, -N, -N; 1-2N, 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^(2m-1) Binomial[2m, m]^2 HypergeometricPFQ[{-m, -m, -m}, {1-2m, 1/2-m}, 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}]
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CROSSREFS
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Related to numbers of walks returning to origin, W_n, on square lattice (A002894).
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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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