A264608 Degeneracies of entanglement witness eigenstates for spin 3 particles.

1, 0, 1, 1, 7, 31, 175, 981, 5719, 33922, 204687, 1251460, 7737807, 48297536, 303922983, 1926038492, 12281450455, 78741558512, 507301771543, 3282586312161, 21323849229781, 139012437340660, 909161626641121, 5963576112550771, 39223341189188339, 258619428254117476, 1709124801693650075

Offset: 0

Author

N. J. A. Sloane, Nov 24 2015

Keywords

• nonn

Examples

			A(x) = 1 + x^2 + x^3 + 7*x^4 + 31*x^5 + 175*x^6 + 981*x^7 + ...
		

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.
• T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, arXiv:1607.05849 [hep-th], 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, this sequence, A272392, A272393, A272394, A272395.

Cf. A348210 (column k=3).

Programs

Mathematica

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(64t^3-80t^2+24t-1)^n, {t, 0, 1}] (* Thomas Curtright, Jun 23 2016 *)
a[n_]:= c[0, n, 3]-c[1, n, 3]; 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, 3]; 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 = 3;
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

PARI

seq(N) = {
  my(a = vector(N), s); a[2]=1; a[3]=1; a[4]=7; a[5]=31;
  for (n=6, N, s = ((n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*(37*n - 38)*a[n-1] + 7*(n-1)*(2*n - 3)*(3*n - 1)*(92*n^3 - 404*n^2 + 509*n - 150)*a[n-2] - 49*(n-2)*(n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*a[n-3] - 343*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a[n-4]);
    a[n] = s/(3*n*(2*n - 5)*(2*n - 3)*(3*n - 4)*(3*n - 1)*(3*n + 1)));
  concat(1,a);
};
seq(26) \\ Gheorghe Coserea, Aug 07 2018

Formula

a(n) ~ (1/8^(3/2))*7^n/(sqrt(Pi)*n^(3/2)) * (1-27/(32*n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 26 2016

D-finite with recurrence 3*n*(2*n - 5)*(2*n - 3)*(3*n - 4)*(3*n - 1)*(3*n + 1)*a(n) = (n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*(37*n - 38)*a(n-1) + 7*(n-1)*(2*n - 3)*(3*n - 1)*(92*n^3 - 404*n^2 + 509*n - 150)*a(n-2) - 49*(n-2)*(n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*a(n-3) - 343*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a(n-4). - Vaclav Kotesovec, Jun 24 2016

a(n) = (1/Pi)*int((sin(7x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - Thomas Curtright, Jun 24 2016

From Gheorghe Coserea, Aug 07 2018: (Start)

G.f. y=A(x) satisfies:

0 = x^3*(x + 1)^4*(49*x^2 - 14*x - 27)^2*y^8 + 2*x^3*(x + 1)^3*(35*x + 23)*(49*x^2 - 14*x - 27)*y^6 + x^2*(x + 1)^2*(1421*x^3 + 1652*x^2 + 393*x - 54)*y^4 + x*(x + 1)*(147*x^3 + 175*x^2 + 51*x - 1)*y^2 + x*(2*x + 1)^2.

0 = x^2*(x + 1)*(7*x - 1)*(7*x + 1)*(49*x^2 - 70*x + 5)*(49*x^2 - 14*x - 27)*y''' + x*(1058841*x^7 - 1092455*x^6 - 1212505*x^5 + 627347*x^4 + 222999*x^3 - 6657*x^2 - 5015*x + 405)*y'' + 2*(1058841*x^7 - 1428595*x^6 - 725102*x^5 + 224322*x^4 + 24157*x^3 + 6909*x^2 - 720*x + 60)*y' + 14*x*(50421*x^5 - 84035*x^4 - 19894*x^3 - 2058*x^2 + 665*x - 75)*y.

(End)

Extensions

More terms from Gheorghe Coserea, Apr 28 2016