A120069 Denominators of partial sums of a convergent series involving scaled Catalan numbers A000108.

1, 2, 16, 32, 128, 256, 4096, 8192, 32768, 65536, 524288, 1048576, 4194304, 8388608, 268435456, 536870912, 2147483648, 4294967296, 34359738368, 68719476736, 274877906944, 549755813888, 8796093022208

Offset: 1

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

For the corresponding numerator sequence see A119951.

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).

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).

The partial sums r(n):=Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).

For the rationals r(n) see the W. Lang link under A119951.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000108, A119951 (numerators).

Programs

Mathematica

Denominator[Table[Sum[CatalanNumber[k]/2^(2*(k - 1)), {k, 1, n}], {n, 1, 50}]] (* G. C. Greubel, Feb 08 2017 *)

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

Formula

a(n) = denominator(r(n)) with the rationals r(n) defined above.

Extensions

First comment corrected by Harvey P. Dale, Oct 09 2017