A272392 Degeneracies of entanglement witness eigenstates for 2n spin 7/2 irreducible representations.

1, 1, 8, 260, 11096, 518498, 25593128, 1312660700, 69270071480, 3736677346685, 205125498479384, 11421904528488264, 643564228586076344, 36624864117451994600, 2102142593641513473240, 121548403269918189484872, 7073453049221266117909752, 413976401197504361048673896

Offset: 0

Author

Gheorghe Coserea, Apr 28 2016

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..200
• Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
• T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.

Crossrefs

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, A272391, A264608, this sequence, A272393, A272394, A272395.

Programs

Mathematica

a[n_]:= c[0, 2*n, 7/2]-c[1, 2*n, 7/2]; c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s], {k, 0, Min[n, Floor[(j + n*s)/(2*s + 1)]]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)

PARI

N = 44; S = 7/2;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0)) \\ Gheorghe Coserea, Apr 28 2016

Formula

a(n) ~ (2*sqrt(42)/441)*8^(2*n)/(sqrt(Pi)*(2*n)^(3/2)) * (1-23/(56*n)+O(1/n^2)). - Thomas Curtright and Cosmas Zachos, Jun 17 2016, updated Jul 26 2016

Recurrence: 7*n*(3*n - 7)*(3*n - 4)*(7*n - 19)*(7*n - 12)*(7*n - 11)*(7*n - 5)*(7*n - 4)*(7*n - 3)*(7*n - 2)*(7*n - 1)*(7*n + 1)*(9*n - 29)*(9*n - 20)*(9*n - 13)*(9*n - 11)*a(n) = 32*(2*n - 1)*(3*n - 7)*(7*n - 19)*(7*n - 12)*(7*n - 11)*(9*n - 29)*(9*n - 20)*(218437803*n^9 - 1510747767*n^8 + 4498401903*n^7 - 7551222032*n^6 + 7855986297*n^5 - 5239178603*n^4 + 2233354977*n^3 - 584916638*n^2 + 85090380*n - 5216400)*a(n-1) - 6144*(n-1)*(2*n - 3)*(2*n - 1)*(7*n - 19)*(9*n - 29)*(9*n - 2)*(460622295*n^10 - 5800755303*n^9 + 31804940376*n^8 - 99676215732*n^7 + 197077947989*n^6 - 255958437117*n^5 + 220361564054*n^4 - 123775781978*n^3 + 43301190686*n^2 - 8505466270*n + 711711000)*a(n-2) + 262144*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 1)*(7*n - 5)*(9*n - 11)*(9*n - 2)*(2988657*n^7 - 36693972*n^6 + 183168228*n^5 - 477680566*n^4 + 695101884*n^3 - 556549424*n^2 + 223584828*n - 34734735)*a(n-3) - 16777216*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 4)*(3*n - 1)*(7*n - 12)*(7*n - 5)*(7*n - 4)*(9*n - 20)*(9*n - 11)*(9*n - 4)*(9*n - 2)*a(n-4). - Vaclav Kotesovec, Jun 24 2016

a(n) = (1/Pi)*int((sin(8x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016