A308612
Moments of the ternary Cantor measure (numerators).
Original entry on oeis.org
1, 1, 3, 5, 87, 31, 10215, 2675, 2030721, 3791353, 21043039755, 3617048975, 456510966890031, 66989072882759, 4380916942893971361, 8410761713598485675, 4355410489470724905492213, 2471070675390828342358441, 8410576873515817323688553597445
Offset: 0
- 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
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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[Numerator[a[i]], {i, 0, 19}] (* Amiram Eldar, Aug 03 2019 *)
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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.numerator() for p in am(15, (1/2, 0, 1/2))]
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
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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 *)
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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]
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
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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 *)
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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]]
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