login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A288455 Chebyshev coefficients of density of states of BCC lattice. 8
1, -48, 1728, -79872, 4058112, -216956928, 11977752576, -676117610496, 38792847949824, -2253773963526144, 132241430641901568, -7821943674035503104, 465750331610495975424, -27888626411947306254336, 1677958399935741979262976, -101375476324084742212288512, 6146869366762959307806867456 (list; graph; refs; listen; history; text; internal format)
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 body-centered cubic 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.

For the bcc lattice (z=8), the even Chebyshev coefficients can be expressed in closed form in terms of the hypergeometric function pFq, as z^{2N} g_{2N} = (1 + delta_N) * 2^(2N-1) Binomial(2N,N)^3 * 4F3 (-N, -N, -N, -N; 1-2N, 1/2-N, 1/2-N; 1).

LINKS

Table of n, a(n) for n=0..16.

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

MATHEMATICA

zng[n_] :=

  If[OddQ[n], 0,

  (1 + KroneckerDelta[m]) 2^(2 m - 1) *

  Binomial[2m, m]^3 HypergeometricPFQ[{-m, -m, -m, -m}, {1-2m, 1/2-m, 1/2-m}, 1] /. m -> n/2];

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

Wchain[n_] := If[OddQ[n], 0, Binomial[n, n/2]];

Wbcc[n_] := Wchain[n]^3;

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

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

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

CROSSREFS

Related to numbers of walks returning to origin, W_n, on BCC lattice (A002897).

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

Sequence in context: A076003 A008845 A273627 * A231450 A152068 A290404

Adjacent sequences:  A288452 A288453 A288454 * A288456 A288457 A288458

KEYWORD

sign

AUTHOR

Yen-Lee Loh, Jun 16 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 15 22:02 EST 2019. Contains 330012 sequences. (Running on oeis4.)