A308615 -id:A308615 - cp's OEIS frontend

1, 1, 7, 205, 10241, 26601785, 144273569311, 8432005793267, 85813777224887042933, 41391682933691854767291415, 279988393358814530594186727509023, 4597481350195941947735138659876438945979, 137236498421201646022141003769649699705393990756253

Offset: 0

Alexander Riasanovsky, Jun 10 2019

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), ....
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.

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.

Matching denominators are A308615. Shifted version of A308612 and A308613.

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 *)

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]

More terms from Amiram Eldar, Aug 03 2019

cp's OEIS Frontend

A308614 Numerators of the even shifted moments of the ternary Cantor measure.

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