A095845 - OEIS

%I #35 Jan 14 2025 06:01:35

%S 1,2,10,5,230,46,874,8740,1673710,1673710,513828970,256914485,

%T 631290272542,3156451362710,15513958447719650,12411166758175720,

%U 305013731457236950790,305013731457236950790,119935974414957427604889850,3156209853025195463286575

%N Denominator of the integral of the n-th power of the Cantor function.

%H Amiram Eldar, <a href="/A095845/b095845.txt">Table of n, a(n) for n = 0..117</a>

%H E. A. Gorin and B. N. Kukushkin, <a href="https://doi.org/10.1090/S1061-0022-04-00817-9">Integrals related to the Cantor function</a>, St. Petersburg Math. J., 15, 449-468, 2004.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CantorFunction.html">Cantor Function</a>.

%F The integral, a rational number, is given by J(n) = 1/(n+1) - Sum_{k = 1..floor(n/2)} binomial(n,2*k)*(2^(2*k-1)-1)*bernoulli(2*k)/((3*2^(2*k-1)-1)*(n-2*k+1)). - _Emeric Deutsch_, Feb 22 2005

%F Note that the Cantor function C(x) satisfies C(x) = C(3*x)/2 for x in [0,1/3], 1/2 for x in [1/3,2/3] and (1+C(3*x-2))/2 for x in [2/3,1]. Integrating both sides yields J(n) = (1 + Sum_{k=0..n-1} binomial(n,k)*J(k))/(3*2^n - 2) with J(0) = 1, where J(n) := Integral_{x=0..1} (C(x))^n dx. - _Jianing Song_, Nov 19 2023

%e 1, 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, 611/8740, 97653/1673710, ...

%p seq(denom(1/(n+1)-sum(binomial(n,2*k)*(2^(2*k-1)-1)*bernoulli(2*k)/(3*2^(2*k-1)-1)/(n-2*k+1),k = 1 .. floor(1/2*n))),n=1..17); # _Emeric Deutsch_, Feb 22 2005

%t a[n_] := Denominator[1/(n + 1) - Sum[(Binomial[n, 2*k]*Floor[2^(2*k - 1) - 1]*BernoulliB[2*k])/Floor[(3*2^(2*k - 1) - 1)*(-2*k + n + 1)], {k, 1, Floor[n/2]}]]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Oct 23 2012, after _Emeric Deutsch_ *)

%t f[0] = 1; f[1] = 1/2; f[n_] := f[n] = (1/(3*2^n - 2))*(2 + Sum[Binomial[n, k]*f[k], {k, 1, n - 1}]); Denominator[Array[f, 20, 0]] (* _Amiram Eldar_, Jan 26 2024 *)

%Y Cf. A095844 (numerators).

%K nonn,frac,changed

%O 0,2

%A _Eric W. Weisstein_, Jun 08 2004