OFFSET
0,3
COMMENTS
Due to the symmetry of the measure mu with respect to x=1/2 and the parity of the polynomial (x-1/2)^k about the line x=1/2, every odd entry is 0 and is thus omitted.
The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2. Here, for j in {0,1,2}, phi_j:[0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3. For any nonnegative integer k, we define the k-th shifted moment J(k) to be the integral of (x-1/2)^k with respect to mu. The described sequence J(0), J(1), J(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of numerators of J(0), J(2), J(4), ....
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..48
Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.
MATHEMATICA
f[0] = 1; f[n_] := f[n] = Sum[Binomial[n, j]*2^(n - j - 1)*f[j], {j, 0, n - 1}]/(3^n - 1); a[n_] := Sum[Binomial[n, j]*f[j]*(-1/2)^(n - j), {j, 0, n}]; Table[Numerator[a[i]], {i, 0, 24, 2}] (* Amiram Eldar, Aug 03 2019 *)
PROG
(Sage)
moms = [1]
for k in [1..15]:
s = 0
for j in [0..k-1]:
s += binomial(k, j)*2^(k-j)*moms[j]/2
s /= (3^k-1)
moms.append(s)
var('x')
shmoms = []
for k in [0..15]:
p = (x-1/2)^k
p = p.expand()
s = 0
for m in [0..k]:
s += moms[m]*p.coefficient(x, m)
shmoms.append(s)
[p.numerator() for p in shmoms if p]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Alexander Riasanovsky, Jun 10 2019
EXTENSIONS
More terms from Amiram Eldar, Aug 03 2019
STATUS
approved