A272395 Degeneracies of entanglement witness eigenstates for n spin 5 irreducible representations.

1, 0, 1, 1, 11, 76, 671, 5916, 54131, 504316, 4779291, 45898975, 445798221, 4371237794, 43213522209, 430241859971, 4310236148075, 43417944574136, 439495074016427, 4468208369691396, 45605656313488271, 467140985042718910

Offset: 0

Author

Gheorghe Coserea, Apr 28 2016

Keywords

• nonn

Comments

The Mathematica formula for a(n) as the difference of two generalized binomial coefficients is adapted from the Appendix of the Mendonça link. - Thomas Curtright, Jul 27 2016

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..401
• 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.
• Vaclav Kotesovec, Recurrence (of order 6)
• J. R. G. Mendonça, Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite and related graphs, Journal of Physics A: Mathematical and Theoretical 46:29 (2013) 295001. arXiv:1207.4106 [cond-mat.stat-mech], 2012-2013.

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, A272392, A272393, A272394, this sequence.

Programs

Mathematica

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(1024t^5-2304t^4+1792t^3-560t^2 +60t-1)^n, {t, 0, 1}] (* Thomas Curtright, Jun 24 2016 *)
a[n_]:= c[0, n, 5]-c[1, n, 5]
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, Floor[(j + n*s)/(2*s + 1)]}] (* Thomas Curtright, Jul 26 2016 *)
a[n_]:= mult[0, n, 5]
mult[j_,n_,s_]:=Sum[(-1)^(k+1)*Binomial[n,k]*Binomial[n*s+j-(2*s+1)*k+n- 1,n*s+j-(2*s+1)*k+1],{k,0,Floor[(n*s+j+1)/(2*s+1)]}] (* Thomas Curtright, Jun 14 2017 *)

PARI

N = 22; S = 5;
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))

PARI

c(j, n) = sum(k=0, min((j + 5*n)\11, n), (-1)^k*binomial(n, k)*binomial(j - 11*k + n + 5*n - 1, j - 11*k + n*5))
a(n)=c(0, n)-c(1, n) \\ Charles R Greathouse IV, Jul 28 2016; adapted from Curtright's Mathematica code

Formula

a(n)=(1/Pi)*int((sin(11x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - Thomas Curtright, Jun 24 2016

a(n) ~ (1/20)^(3/2)*11^n/(sqrt(Pi)*n^(3/2))(1-63/(80n)+O(1/n^2)). - Thomas Curtright, Jul 26 2016