A308612 -id:A308612 - cp's OEIS frontend

A308613 Moments of the ternary Cantor measure (denominators).

1, 2, 8, 16, 320, 128, 46592, 13312, 10915840, 21831680, 128911704064, 23438491648, 3114038000353280, 479082769285120, 32734822212030169088, 65469644424060338176, 35228168150276083007094784, 20722451853103578239467520, 72984567358962659964369885986816

Offset: 0

Alexander Riasanovsky, Jun 10 2019

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 moment to be I(k) to be the integral of x^k with respect to mu.  The described sequence I(0), I(1), I(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of denominators of I(0), I(1), I(2), ....
For the purpose of computing I(k), we note the following recurrence relation: I(0) = 1 and for all positive k, I(k) = (1/(3^k-1))*((1/2) * Sum_{j=0..k-1} binomial(k, j) + (1/2) * Sum_{j=0..k-1} binomial(k, j) * 2^(k-j) * I(j)).
More generally, for any N-dimensional nonnegative vector alpha = (alpha_0, ..., alpha_{N-1}) whose entries sum to 1, there exists a unique Borel measure mu = mu^{alpha} on [0,1] so that for any measurable set E, the following identity holds: mu(E) = Sum_{k=0..N-1} alpha_k * mu(phi_k(E)).  Here, for j in {0, 1, ..., N-1}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/N.  Defining I(k) to be the integral of x^k with respect to mu, the following recurrence relation holds: I(0) = 1 and for all positive k, I(k) = (1/(N^k-1)) * Sum_{n=0..N-1} alpha_n * Sum_{j=0..k-1} binomial(k, j) * n^(k-j)*I(j).

Michael De Vlieger, Table of n, a(n) for n = 0..97
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Michael G. Rose, Expectations on fractal sets, Applied Mathematics and Computation, Vol. 220 (2013), pp. 695-721, alternative link.
Steven N. Harding, Alexander W. N. Riasanovsky, Moments of the weighted Cantor measures, arXiv:1908.05358 [math.FA], 2019.
Math Stack Exchange, Integrating f(x) = x for x in C, the Cantor set, with respect to a certain measure

Cf. A308612 (numerators).

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, j]*2^(n - j - 1)*a[j], {j, 0, n - 1}]/(3^n - 1); Table[Denominator[a[i]], {i, 0, 19}] (* Amiram Eldar, Aug 03 2019 *)

def am(m, alpha):
    N = len(alpha)
    am = [1]
    for a in [1..m]:
        mm = 0
        for k in [0..N-1]:
            for r in [0..a-1]:
                mm += alpha[k]*binomial(a, r)*k^(a-r)*am[r]
        mm /= (N^a-1)
        am.append(mm)
    return am
[p.denominator() for p in am(15, (1/2, 0, 1/2))]

More terms from Amiram Eldar, Aug 03 2019

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

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 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.

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 numerators are A308614. 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[Denominator[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)
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]]

More terms from Amiram Eldar, Aug 03 2019

cp's OEIS Frontend

A308613 Moments of the ternary Cantor measure (denominators).

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

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

Sage

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Extensions