%I #16 Jun 19 2017 15:15:34

%S 1,-48,1344,-24576,218112,-688128,926416896,-95932121088,

%T 5186228846592,-154060166529024,1455620351852544,-29436202608230400,

%U 17834604768232734720,-1968810407797802926080,114581075578951670169600,-3629224301781687956668416,33517817437575659447648256,-1040884075746436707891806208

%N Chebyshev coefficients of density of states of 4D hypercubic lattice.

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

%H 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 Wdia[n_] := If[OddQ[n], 0,

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

%t Whcub[n_] := Binomial[n, n/2] Wdia[n];

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

%t zng[n_] := Sum[ank[n, k]*8^(n-k)*Whcub[k], {k, 0, n}];

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

%Y Related to numbers of walks returning to origin, W_n, on hypercubic lattice (A039699).

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

%K sign

%O 0,2

%A _Yen-Lee Loh_, Jun 16 2017