

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
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OFFSET

1,2


COMMENTS

For the corresponding numerator sequence see A119951.
The series s:=sum(C(k)/2^(2*(k1)),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*(k1)) 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*(k1)),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^(n1) 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*k2)))), ", ")) \\ 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



