A272394 - OEIS

%I #53 Jul 28 2016 00:52:06

%S 1,1,10,505,33670,2457190,189442252,15177798415,1251216059950,

%T 105443928375598,9043123211156440,786701771691580227,

%U 69253844083218535300,6157639918607211895000,552193624489443516667344,49885368826043082235592687

%N Degeneracies of entanglement witness eigenstates for 2n spin 9/2 irreducible representations.

%H Gheorghe Coserea, <a href="/A272394/b272394.txt">Table of n, a(n) for n = 0..200</a>

%H 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 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 Vaclav Kotesovec, <a href="/A272394/a272394.txt">Recurrence (of order 5)</a>

%F a(n) = (1/Pi)*int((sin(10x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - _Thomas Curtright_, Jun 24 2016

%F a(n) ~ (2*sqrt(66)/1089)*10^(2n)/(sqrt(Pi)*(2n)^(3/2))(1-35/(88n) + O(1/n^2)). - _Thomas Curtright_, Jul 26 2016

%t a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(4t)^n*((16t^2-20t+5)((4t-1)^2-4t))^(2n), {t, 0, 1}] (* _Thomas Curtright_, Jun 24 2016 *)

%t a[n_]:= c[0, 2 n, 9/2]-c[1, 2 n, 9/2]

%t 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)]]}] (* _Thomas Curtright_, Jul 26 2016 *)

%o (PARI)

%o N = 33; S = 9/2;

%o M = matrix(N+1, N*numerator(S)+1);

%o Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };

%o Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };

%o Minit() = {

%o   my(step = 1/denominator(S));

%o   Mset(0, 0, 1);

%o   for (n = 1, N, forstep (j = 0, n*S, step,

%o      my(acc = 0);

%o      for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));

%o      Mset(n, j, acc)));

%o };

%o Minit();

%o vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0))

%Y 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, A272393, this sequence, A272395.

%K nonn

%O 0,3

%A _Gheorghe Coserea_, Apr 28 2016