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A288458 Chebyshev coefficients of density of states of cubic lattice. 8
1, -24, 288, -2688, -32256, 2820096, -95035392, 1972076544, -9841803264, -1288894414848, 70351960670208, -2164060518875136, 36664809432809472, 365875642245316608, -55960058736918134784, 2436570173137823465472, -64272155689216515244032, 664295705652718630600704, 35692460661517822602510336 (list; graph; refs; listen; history; text; internal format)



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 simple cubic lattice (z=6), 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.

The author was unable to obtain a closed form for z^n g_n.


Table of n, a(n) for n=0..18.

Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.


Whon[n_] := If[OddQ[n], 0,

   Sum[Binomial[n/2, j]^2 Binomial[2j, j], {j, 0, n/2}]];

Wcub[n_] := Binomial[n, n/2] Whon[n];

ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];

zng[n_] := Sum[ank[n, k]*6^(n-k)*Wcub[k], {k, 0, n}];

Table[zng[n], {n, 0, 50}]


Related to numbers of walks returning to origin, W_n, on cubic lattice (A002896).

See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.

Sequence in context: A282154 A035707 A035475 * A042110 A282993 A295272

Adjacent sequences:  A288455 A288456 A288457 * A288459 A288460 A288461




Yen-Lee Loh, Jun 16 2017



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Last modified October 18 06:01 EDT 2021. Contains 348065 sequences. (Running on oeis4.)