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A264608 Degeneracies of entanglement witness eigenstates for spin 3 particles.

%I A264608 #68 Oct 12 2021 07:54:22
%S A264608 1,0,1,1,7,31,175,981,5719,33922,204687,1251460,7737807,48297536,
%T A264608 303922983,1926038492,12281450455,78741558512,507301771543,
%U A264608 3282586312161,21323849229781,139012437340660,909161626641121,5963576112550771,39223341189188339,258619428254117476,1709124801693650075
%N A264608 Degeneracies of entanglement witness eigenstates for spin 3 particles.
%H A264608 Gheorghe Coserea, <a href="/A264608/b264608.txt">Table of n, a(n) for n = 0..401</a>
%H A264608 Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, <a href="http://arxiv.org/abs/1511.06623">From Entanglement Witness to Generalized Catalan Numbers</a>, arXiv:1511.06623 [quant-ph], 2015.
%H A264608 T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, <a href="https://hal.archives-ouvertes.fr/hal-01345527">Spin Multiplicities</a>, hal-01345527, 2016.
%H A264608 T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, <a href="https://arxiv.org/abs/1607.05849">Spin Multiplicities</a>, arXiv:1607.05849 [hep-th], 2016.
%F A264608 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
%F A264608 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
%F A264608 a(n) = (1/Pi)*int((sin(7x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - _Thomas Curtright_, Jun 24 2016
%F A264608 From _Gheorghe Coserea_, Aug 07 2018: (Start)
%F A264608 G.f. y=A(x) satisfies:
%F A264608 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.
%F A264608 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.
%F A264608 (End)
%e A264608 A(x) = 1 + x^2 + x^3 + 7*x^4 + 31*x^5 + 175*x^6 + 981*x^7 + ...
%t A264608 a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(64t^3-80t^2+24t-1)^n, {t, 0, 1}] (* _Thomas Curtright_, Jun 23 2016 *)
%t A264608 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 *)
%t A264608 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 *)
%o A264608 (PARI)
%o A264608 N = 26; S = 3;
%o A264608 M = matrix(N+1, N*numerator(S)+1);
%o A264608 Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
%o A264608 Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
%o A264608 Minit() = {
%o A264608   my(step = 1/denominator(S));
%o A264608   Mset(0, 0, 1);
%o A264608   for (n = 1, N, forstep (j = 0, n*S, step,
%o A264608      my(acc = 0);
%o A264608      for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
%o A264608      Mset(n, j, acc)));
%o A264608 };
%o A264608 Minit();
%o A264608 vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0)) \\ _Gheorghe Coserea_, Apr 28 2016
%o A264608 (PARI)
%o A264608 seq(N) = {
%o A264608   my(a = vector(N), s); a[2]=1; a[3]=1; a[4]=7; a[5]=31;
%o A264608   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]);
%o A264608     a[n] = s/(3*n*(2*n - 5)*(2*n - 3)*(3*n - 4)*(3*n - 1)*(3*n + 1)));
%o A264608   concat(1,a);
%o A264608 };
%o A264608 seq(26) \\ _Gheorghe Coserea_, Aug 07 2018
%Y A264608 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.
%Y A264608 Cf. A348210 (column k=3).
%K A264608 nonn
%O A264608 0,5
%A A264608 _N. J. A. Sloane_, Nov 24 2015
%E A264608 More terms from _Gheorghe Coserea_, Apr 28 2016