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%I #64 Feb 08 2021 06:38:47
%S 1,1,4,34,364,4269,52844,679172,8976188,121223668,1665558544,
%T 23207619274,327167316436,4657884819670,66875794530120,
%U 967202289590280,14077773784645980,206058395118133932,3031188276557963312,44789055557553810152
%N Degeneracies of entanglement witness eigenstates for spin 3/2 particles.
%H Gheorghe Coserea, <a href="/A264607/b264607.txt">Table of n, a(n) for n = 0..200</a>
%H Hacène Belbachir, Oussama Igueroufa, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=48">Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
%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.
%F 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
%F 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
%F a(n) = (1/Pi)*int((sin(4x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - _Thomas Curtright_, Jun 24 2016
%F a(n) = Catalan(3*n)*2F1(-1-3*n,-2*n;1/2-3*n;1/2). - _Benedict W. J. Irwin_, Sep 27 2016
%t 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 *)
%t 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 *)
%t Table[CatalanNumber[3 n]Hypergeometric2F1[-1-3n,-2n,1/2-3n,1/2],{n,0,20}] (* _Benedict W. J. Irwin_, Sep 27 2016 *)
%o (PARI)
%o N = 44; S = 3/2;
%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)) \\ _Gheorghe Coserea_, Apr 28 2016
%Y 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.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Nov 24 2015
%E More terms from _Gheorghe Coserea_, Apr 28 2016
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