cp's OEIS Frontend

A288459 Chebyshev coefficients of density of states of 4D hypercubic lattice.

%I A288459 #16 Jun 19 2017 15:15:34
%S A288459 1,-48,1344,-24576,218112,-688128,926416896,-95932121088,
%T A288459 5186228846592,-154060166529024,1455620351852544,-29436202608230400,
%U A288459 17834604768232734720,-1968810407797802926080,114581075578951670169600,-3629224301781687956668416,33517817437575659447648256,-1040884075746436707891806208
%N A288459 Chebyshev coefficients of density of states of 4D hypercubic lattice.
%C A288459 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 four-dimensional hypercubic 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 A288459 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 A288459 The author was unable to obtain a closed form for z^n g_n.
%H A288459 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 A288459 Wdia[n_] := If[OddQ[n], 0,
%t A288459    Sum[Binomial[n/2,j]^2 Binomial[2j,j] Binomial[n-2j, n/2-j], {j, 0, n/2}]];
%t A288459 Whcub[n_] := Binomial[n, n/2] Wdia[n];
%t A288459 ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
%t A288459 zng[n_] := Sum[ank[n, k]*8^(n-k)*Whcub[k], {k, 0, n}];
%t A288459 Table[zng[n], {n,0,50}]
%Y A288459 Related to numbers of walks returning to origin, W_n, on hypercubic lattice (A039699).
%Y A288459 See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.
%K A288459 sign
%O A288459 0,2
%A A288459 _Yen-Lee Loh_, Jun 16 2017