OFFSET
1,1
COMMENTS
T(n, k) is the number of k-th roots of unity as eigenvalues of the quantum operator O for a free Motzkin spin chain of length n. For k = 1, it gives the correct result if one excludes the eigenvalue 2.
For the definitions of both a free Motzkin spin chain and the quantum operator O, see Hao et al.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Kun Hao, Olof Salberger, and Vladimir Korepin, Can a spin chain relate combinatorics to number theory?, arXiv:2202.07647 [quant-ph], 2022. See pp. 9-10.
EXAMPLE
Triangle begins:
2;
6, 2;
14, 6, 6;
30, 14, 24, 12;
62, 30, 60, 60, 30;
126, 62, 126, 180, 180, 54;
254, 126, 252, 420, 630, 378, 126;
...
MATHEMATICA
g[n_]:= DivisorSum[n, (2^#)*MoebiusMu[n/#]&]; binomSum[n_, k_]:=Sum[Binomial[n, i], {i, k, n, k}]; T[n_, k_]:=g[k]*binomSum[n, k]; (* See p. 9 in Hao et al. *)
Flatten[Table[T[n, k], {n, 10}, {k, n}]]
PROG
(PARI) T(n, k) = sumdiv(k, d, moebius(d)*2^(k/d))*sum(m=1, n\k, binomial(n, k*m)) \\ Andrew Howroyd, Feb 21 2022
CROSSREFS
AUTHOR
Stefano Spezia, Feb 21 2022
STATUS
approved