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

1, -48, 1344, -24576, 218112, -688128, 926416896, -95932121088, 5186228846592, -154060166529024, 1455620351852544, -29436202608230400, 17834604768232734720, -1968810407797802926080, 114581075578951670169600, -3629224301781687956668416, 33517817437575659447648256, -1040884075746436707891806208

Offset

0,2

Comments

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.

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.

Links

Table of n, a(n) for n=0..17.

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

Mathematica

Wdia[n_] := If[OddQ[n], 0,
   Sum[Binomial[n/2, j]^2 Binomial[2j, j] Binomial[n-2j, n/2-j], {j, 0, n/2}]];
Whcub[n_] := Binomial[n, n/2] Wdia[n];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*8^(n-k)*Whcub[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]

Crossrefs

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

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

Sequence in context: A141407 A004341 A222199 * A173155 A174113 A089272

Adjacent sequences: A288456 A288457 A288458 * A288460 A288461 A288462

Keyword

sign

Author

Yen-Lee Loh, Jun 16 2017

Status

approved