A272393 Degeneracies of entanglement witness eigenstates for n spin 4 irreducible representations.

1, 0, 1, 1, 9, 51, 369, 2661, 19929, 151936, 1178289, 9259812, 73593729, 590475744, 4776464121, 38912018796, 318971849625, 2629040965776, 21774894337449, 181136924953317, 1512731101731499, 12678230972826340, 106600213003114719

Offset: 0

Author

Gheorghe Coserea, Apr 28 2016

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..401
• Eliahu Cohen, Tobias Hansen, and 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, A272392, this sequence, A272394, A272395.

Cf. A348210 (column k=4).

Programs

Mathematica

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*((4t-1)(4t(4t-3)^2-1))^n, {t, 0, 1}] (* Thomas Curtright, Jun 24 2016 *)
a[n_]:= c[0, n, 4]-c[1, n, 4]; 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)]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)
a[n_]:= mult[0, n, 4]
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 = 26; S = 4;
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))

Formula

a(n) ~ (3/40)^(3/2)*9^n/(sqrt(Pi)*n^(3/2)) * (1-129/(160*n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 26 2016

D-finite with recurrence 8*n*(2*n - 5)*(2*n - 3)*(2*n - 1)*(4*n - 1)*(4*n + 1)*(5*n - 17)*(5*n - 12)*(5*n - 8)*(5*n - 7)*a(n) = (n-1)*(2*n - 5)*(2*n - 3)*(5*n - 17)*(5*n - 12)*(5*n - 3)*(5*n - 2)*(2101*n^3 - 6506*n^2 + 6608*n - 2200)*a(n-1) + 9*(n-1)*(2*n - 5)*(2*n - 1)*(4*n - 1)*(5*n - 17)*(5*n - 8)*(5*n - 7)*(385*n^3 - 1659*n^2 + 2008*n - 492)*a(n-2) - 81*(n-2)*(n-1)*(2*n - 3)*(5*n - 12)*(5*n - 3)*(5*n - 2)*(1020*n^4 - 8088*n^3 + 21761*n^2 - 22557*n + 7264)*a(n-3) - 729*(n-3)*(n-2)*(n-1)*(2*n - 5)*(2*n - 1)*(4*n - 1)*(5*n - 17)*(5*n - 8)*(5*n - 7)*(5*n - 2)*a(n-4) + 6561*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(5*n - 12)*(5*n - 7)*(5*n - 3)*(5*n - 2)*a(n-5). - Vaclav Kotesovec, Jun 24 2016

a(n) = (1/Pi)*int((sin(9x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - Thomas Curtright, Jun 24 2016