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.
%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