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A120069 Denominators of partial sums of a convergent series involving scaled Catalan numbers A000108. 3
1, 2, 16, 32, 128, 256, 4096, 8192, 32768, 65536, 524288, 1048576, 4194304, 8388608, 268435456, 536870912, 2147483648, 4294967296, 34359738368, 68719476736, 274877906944, 549755813888, 8796093022208 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the corresponding numerator sequence see A119951.

The series s:=sum(C(k)/2^(2*(k-1)),k=1..infty), with C(n):=A000108(n) (Catalan numbers) is convergent due to J. L. Raabe's criterion. The value for s is 4 (see A119951).

The asymptotics for C(n)/2^(2*(k-1)) is 4/(sqrt(Pi)*k^(3/2)) (see mathworld). The sum over the asymptotic values from k=1..infinity is (4/sqrt(Pi))*Zeta(3/2) = 5.895499840 (maple10, 10 digits).

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

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

Term 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

FORMULA

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Sequence in context: A056707 A069256 A279034 * A018975 A012696 A012392

Adjacent sequences:  A120066 A120067 A120068 * A120070 A120071 A120072

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

EXTENSIONS

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

STATUS

approved

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Last modified February 22 05:17 EST 2019. Contains 320385 sequences. (Running on oeis4.)