cp's OEIS Frontend

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

%I A272393 #75 Mar 07 2025 01:29:07
%S A272393 1,0,1,1,9,51,369,2661,19929,151936,1178289,9259812,73593729,
%T A272393 590475744,4776464121,38912018796,318971849625,2629040965776,
%U A272393 21774894337449,181136924953317,1512731101731499,12678230972826340,106600213003114719
%N A272393 Degeneracies of entanglement witness eigenstates for n spin 4 irreducible representations.
%H A272393 Gheorghe Coserea, <a href="/A272393/b272393.txt">Table of n, a(n) for n = 0..401</a>
%H A272393 Eliahu Cohen, Tobias Hansen, and 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 A272393 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.
%F A272393 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
%F A272393 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
%F A272393 a(n) = (1/Pi)*int((sin(9x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - _Thomas Curtright_, Jun 24 2016
%t A272393 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 *)
%t A272393 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 *)
%t A272393 a[n_]:= mult[0, n, 4]
%t A272393 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 A272393 (PARI)
%o A272393 N = 26; S = 4;
%o A272393 M = matrix(N+1, N*numerator(S)+1);
%o A272393 Mget(n, j) = { M[1 + n, 1 + j*denominator(S)] };
%o A272393 Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
%o A272393 Minit() = {
%o A272393   my(step = 1/denominator(S));
%o A272393   Mset(0, 0, 1);
%o A272393   for (n = 1, N, forstep (j = 0, n*S, step,
%o A272393      my(acc = 0);
%o A272393      for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
%o A272393      Mset(n, j, acc)));
%o A272393 };
%o A272393 Minit();
%o A272393 vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0))
%Y A272393 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.
%Y A272393 Cf. A348210 (column k=4).
%K A272393 nonn
%O A272393 0,5
%A A272393 _Gheorghe Coserea_, Apr 28 2016