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

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

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), ....

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.

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

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

Sequence in context: A216456 A243144 A234619 * A178022 A157775 A300619

Adjacent sequences: A308611 A308612 A308613 * A308615 A308616 A308617

Keyword

nonn,frac

Author

Alexander Riasanovsky, Jun 10 2019

Extensions

More terms from Amiram Eldar, Aug 03 2019

Status

approved