A351839 - OEIS

%I #33 Jan 07 2024 17:16:32

%S 2,6,2,14,6,6,30,14,24,12,62,30,60,60,30,126,62,126,180,180,54,254,

%T 126,252,420,630,378,126,510,254,504,852,1680,1512,1008,240,1022,510,

%U 1014,1620,3780,4536,4536,2160,504,2046,1022,2040,3060,7590,11340,15120,10800,5040,990

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

%C 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.

%C For the definitions of both a free Motzkin spin chain and the quantum operator O, see Hao et al.

%H Andrew Howroyd, <a href="/A351839/b351839.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%H Kun Hao, Olof Salberger, and Vladimir Korepin, <a href="https://arxiv.org/abs/2202.07647">Can a spin chain relate combinatorics to number theory?</a>, arXiv:2202.07647 [quant-ph], 2022. See pp. 9-10.

%e Triangle begins:

%e     2;

%e     6,   2;

%e    14,   6,   6;

%e    30,  14,  24,  12;

%e    62,  30,  60,  60,  30;

%e   126,  62, 126, 180, 180,  54;

%e   254, 126, 252, 420, 630, 378, 126;

%e   ...

%t 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. *)

%t Flatten[Table[T[n,k],{n,10},{k,n}]]

%o (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

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

%K nonn,tabl,look

%O 1,1

%A _Stefano Spezia_, Feb 21 2022