cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

This page as a plain text file.
%I A308615 #37 Mar 18 2020 05:10:21
%S A308615 1,8,320,46592,10915840,128911704064,3114038000353280,
%T A308615 798410297854394368,35228168150276083007094784,
%U A308615 72984567358962659964369885986816,2104733804502091904066890388853154119680,146449616359318768962787815768964807513279037440
%N A308615 Denominators of the even shifted moments of the ternary Cantor measure.
%C A308615 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.
%C A308615 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), ....
%C A308615 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.
%H A308615 Michael De Vlieger, <a href="/A308615/b308615.txt">Table of n, a(n) for n = 0..48</a>
%H A308615 Steven N. Harding, Alexander W. N. Riasanovsky, <a href="https://arxiv.org/abs/1908.05358">Moments of the weighted Cantor measures</a>, arXiv:1908.05358 [math.FA], 2019.
%t A308615 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 *)
%o A308615 (Sage)
%o A308615 moms = [1]
%o A308615 for k in [1..15]:
%o A308615     s = 0
%o A308615     for j in [0..k-1]:
%o A308615         s += binomial(k, j)*2^(k-j)*moms[j]/2
%o A308615     s /= (3^k-1)
%o A308615     moms.append(s)
%o A308615 x = var('x')
%o A308615 shmoms = []
%o A308615 for k in [0..15]:
%o A308615     p = (x-1/2)^k
%o A308615     p = p.expand()
%o A308615     s = 0
%o A308615     for m in [0..k]:
%o A308615         s += moms[m]*p.coefficient(x, m)
%o A308615     shmoms.append(s)
%o A308615 [p.denominator() for p in shmoms[::2]]
%Y A308615 Matching numerators are A308614.  Shifted version of A308612 and A308613.
%K A308615 nonn,frac
%O A308615 0,2
%A A308615 _Alexander Riasanovsky_, Jun 10 2019
%E A308615 More terms from _Amiram Eldar_, Aug 03 2019