A264607 Degeneracies of entanglement witness eigenstates for spin 3/2 particles.

1, 1, 4, 34, 364, 4269, 52844, 679172, 8976188, 121223668, 1665558544, 23207619274, 327167316436, 4657884819670, 66875794530120, 967202289590280, 14077773784645980, 206058395118133932, 3031188276557963312, 44789055557553810152

Offset: 0

Author

N. J. A. Sloane, Nov 24 2015

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..200
• Hacène Belbachir, Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
• 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, this sequence, A007043, A272391, A264608, A272392, A272393, A272394, A272395.

Programs

Mathematica

a[n_]:= 2/Pi*4^(2*n)*Integrate[Sqrt[1-t]*(2*t-1)^(2*n)*Sqrt[t]^(2*n-1),{t,0,1}] (* Thomas Curtright, Jun 22 2016 *)
a[n_]:= c[0, 2 n, 3/2]-c[1, 2 n, 3/2]; 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 *)
Table[CatalanNumber[3 n]Hypergeometric2F1[-1-3n,-2n,1/2-3n,1/2],{n,0,20}] (* Benedict W. J. Irwin, Sep 27 2016 *)

PARI

N = 44; S = 3/2;
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(10)/25)*4^(2*n)/(sqrt(Pi)*(2*n)^(3/2)) * (1-21/(40*n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 16 2016

D-finite with recurrence: 3*n*(3*n - 1)*(3*n + 1)*(5*n - 7)*a(n) = 8*(2*n - 1)*(145*n^3 - 338*n^2 + 238*n - 51)*a(n-1) - 128*(n-1)*(2*n - 3)*(2*n - 1)*(5*n - 2)*a(n-2). - Vaclav Kotesovec, Jun 24 2016

a(n) = (1/Pi)*int((sin(4x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016

a(n) = Catalan(3*n)*2F1(-1-3*n,-2*n;1/2-3*n;1/2). - Benedict W. J. Irwin, Sep 27 2016

Extensions

More terms from Gheorghe Coserea, Apr 28 2016