A308615 Denominators of the even shifted moments of the ternary Cantor measure.

1, 8, 320, 46592, 10915840, 128911704064, 3114038000353280, 798410297854394368, 35228168150276083007094784, 72984567358962659964369885986816, 2104733804502091904066890388853154119680, 146449616359318768962787815768964807513279037440

Offset: 0

Author

Alexander Riasanovsky, Jun 10 2019

Keywords

• nonn
• frac

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 denominators of J(0), J(2), J(4), ....

For the purpose of computing J(k), we first compute the (unshifted) moments (see A308612 and A308613) which are the integrals of x^k rather than (x-1/2)^k, expand the polynomial (x-1/2)^k, replace each x^m term with the corresponding moment I(m), and simplify.

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.

Crossrefs

Matching numerators are A308614. Shifted version of A308612 and A308613.

Programs

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[Denominator[a[i]], {i, 0, 24, 2}] (* Amiram Eldar, Aug 03 2019 *)

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)
x = 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.denominator() for p in shmoms[::2]]

Extensions

More terms from Amiram Eldar, Aug 03 2019