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A264607
Degeneracies of entanglement witness eigenstates for spin 3/2 particles.
8
1, 1, 4, 34, 364, 4269, 52844, 679172, 8976188, 121223668, 1665558544, 23207619274, 327167316436, 4657884819670, 66875794530120, 967202289590280, 14077773784645980, 206058395118133932, 3031188276557963312, 44789055557553810152
OFFSET
0,3
LINKS
Hacène Belbachir, Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.
FORMULA
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
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
a(n) = (1/Pi)*int((sin(4x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016
a(n) = Catalan(3*n)*2F1(-1-3*n,-2*n;1/2-3*n;1/2). - Benedict W. J. Irwin, Sep 27 2016
MATHEMATICA
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 *)
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 *)
Table[CatalanNumber[3 n]Hypergeometric2F1[-1-3n, -2n, 1/2-3n, 1/2], {n, 0, 20}] (* Benedict W. J. Irwin, Sep 27 2016 *)
PROG
(PARI)
N = 44; S = 3/2;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) = { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
my(step = 1/denominator(S));
Mset(0, 0, 1);
for (n = 1, N, forstep (j = 0, n*S, step,
my(acc = 0);
for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S), 0)) \\ Gheorghe Coserea, Apr 28 2016
CROSSREFS
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.
Sequence in context: A206180 A274344 A199752 * A307941 A084973 A234313
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 24 2015
EXTENSIONS
More terms from Gheorghe Coserea, Apr 28 2016
STATUS
approved