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%I #37 Mar 18 2020 05:16:37
%S 1,1,7,205,10241,26601785,144273569311,8432005793267,
%T 85813777224887042933,41391682933691854767291415,
%U 279988393358814530594186727509023,4597481350195941947735138659876438945979,137236498421201646022141003769649699705393990756253
%N Numerators of the even shifted moments of the ternary Cantor measure.
%C 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 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), ....
%C 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 Michael De Vlieger, <a href="/A308614/b308614.txt">Table of n, a(n) for n = 0..48</a>
%H 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 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 *)
%o (Sage)
%o moms = [1]
%o for k in [1..15]:
%o s = 0
%o for j in [0..k-1]:
%o s += binomial(k, j)*2^(k-j)*moms[j]/2
%o s /= (3^k-1)
%o moms.append(s)
%o var('x')
%o shmoms = []
%o for k in [0..15]:
%o p = (x-1/2)^k
%o p = p.expand()
%o s = 0
%o for m in [0..k]:
%o s += moms[m]*p.coefficient(x, m)
%o shmoms.append(s)
%o [p.numerator() for p in shmoms if p]
%Y Matching denominators are A308615. Shifted version of A308612 and A308613.
%K nonn,frac
%O 0,3
%A _Alexander Riasanovsky_, Jun 10 2019
%E More terms from _Amiram Eldar_, Aug 03 2019
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