|
%I #22 Mar 05 2022 00:28:06
%S 1,2,16,32,128,256,4096,8192,32768,65536,524288,1048576,4194304,
%T 8388608,268435456,536870912,2147483648,4294967296,34359738368,
%U 68719476736,274877906944,549755813888,8796093022208
%N Denominators of partial sums of a convergent series involving scaled Catalan numbers A000108.
%C For the corresponding numerator sequence see A119951.
%C The series s:=Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers) converges by Raabe's test. The value for s is 4 (see A119951).
%C Asymptotically, C(n)/2^(2*(k-1)) ~ 4/(sqrt(Pi)*k^(3/2)) (see Mathworld). The sum of the asymptotic values from k = 1 to infinity is (4/sqrt(Pi))*Zeta(3/2) = 5.895499840 (Maple10, 10 digits).
%C The partial sums r(n):=Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
%C For the rationals r(n) see the W. Lang link under A119951.
%C a(n) appears to be the denominator of Catalan(n)/4^(n-1) but I have no proof of this. - _Groux Roland_, Dec 11 2010
%H G. C. Greubel, <a href="/A120069/b120069.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = denominator(r(n)) with the rationals r(n) defined above.
%t Denominator[Table[Sum[CatalanNumber[k]/2^(2*(k - 1)), {k, 1, n}], {n, 1, 50}]] (* _G. C. Greubel_, Feb 08 2017 *)
%o (PARI) for(n=1,50, print1(denominator(sum(k=1,n, binomial(2*k,k)/((k+1)*2^(2*k-2)))), ", ")) \\ _G. C. Greubel_, Feb 08 2017
%Y Cf. A000108, A119951 (numerators).
%K nonn,easy,frac
%O 1,2
%A _Wolfdieter Lang_, Jul 20 2006
%E First comment corrected by _Harvey P. Dale_, Oct 09 2017
|
