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A288459
Chebyshev coefficients of density of states of 4D hypercubic lattice.
8
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.
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).
Sequence in context: A141407 A004341 A222199 * A173155 A174113 A089272
KEYWORD
sign
AUTHOR
Yen-Lee Loh, Jun 16 2017
STATUS
approved