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%I #62 Nov 03 2020 01:40:44
%S 1,1,6,111,2666,70146,1949156,56267133,1670963202,50720602314,
%T 1566629938776,49080774275121,1555873464248076,49814409137161480,
%U 1608523756282054800,52323002586904505427,1712956041168844662002
%N Degeneracies of entanglement witness eigenstates for 2n spin 5/2 irreducible representations.
%H Gheorghe Coserea, <a href="/A272391/b272391.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 Thomas Curtright, Thomas Van Kortryk, and Cosmas Zachos, <a href="https://hal.archives-ouvertes.fr/hal-01345527">Spin Multiplicities</a>, hal-01345527, 2016.
%F a(n) ~ (6*sqrt(210)/1225)*6^(2*n)/(sqrt(Pi)*(2*n)^(3/2)) * (1-123/(280n)+O(1/n^2)). - _Thomas Curtright_, Jun 17 2016, updated Jul 26 2016
%F Recurrence: 5*n*(5*n - 8)*(5*n - 3)*(5*n - 2)*(5*n - 1)*(5*n + 1)*(7*n - 16)*(7*n - 10)*(7*n - 9)*a(n) = 6*(2*n - 1)*(5*n - 8)*(7*n - 16)*(499359*n^6 - 2314137*n^5 + 4264709*n^4 - 3984323*n^3 + 1983172*n^2 - 496780*n + 48720)*a(n-1) - 864*(n-1)*(2*n - 3)*(2*n - 1)*(7*n - 2)*(25480*n^5 - 160398*n^4 + 375142*n^3 - 401079*n^2 + 192819*n - 33500)*a(n-2) + 31104*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(5*n - 3)*(7*n - 9)*(7*n - 3)*(7*n - 2)*a(n-3). - _Vaclav Kotesovec_, Jun 24 2016
%F a(n) = (1/Pi)*int((sin(6x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - _Thomas Curtright_, Jun 24 2016
%t a[n_] := 2/Pi * 2^(2 * n) * Integrate[Sqrt[1 - t] * ((4 * t - 1)(4 * t - 3))^(2 * n) * Sqrt[t]^(2 * n - 1), {t, 0, 1}] (* _Thomas Curtright_, Jun 23 2016 *)
%t a[n_] := c[0, 2 n, 5/2] - c[1, 2 n, 5/2]; 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 *)
%o (PARI)
%o N = 34; S = 5/2;
%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, this sequence, A264608, A272392, A272393, A272394, A272395.
%K nonn
%O 0,3
%A _Gheorghe Coserea_, Apr 28 2016
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