OFFSET
0,3
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..200
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.
Vaclav Kotesovec, Recurrence (of order 5)
FORMULA
a(n) = (1/Pi)*int((sin(10x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016
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
MATHEMATICA
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 *)
a[n_]:= c[0, 2 n, 9/2]-c[1, 2 n, 9/2]
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)]]}] (* Thomas Curtright, Jul 26 2016 *)
PROG
(PARI)
N = 33; S = 9/2;
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))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Apr 28 2016
STATUS
approved