A351839 Triangle read by rows: T(n, k) = A027375(n)*Sum_{m=1..floor(n/k)} binomial(n, k*m).

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, 510, 254, 504, 852, 1680, 1512, 1008, 240, 1022, 510, 1014, 1620, 3780, 4536, 4536, 2160, 504, 2046, 1022, 2040, 3060, 7590, 11340, 15120, 10800, 5040, 990

Offset: 1

Author

Stefano Spezia, Feb 21 2022

Keywords

• nonn
• tabl
• look

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.

Examples

			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;
  ...
		

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.

Crossrefs

Cf. A000918 (k = 2), A007318, A024023 (row sums), A027375 (leading diagonal), A095121 (k = 1).

Programs

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}]]

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