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A120069
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Denominators of partial sums of a convergent series involving scaled Catalan numbers A000108.
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3
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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
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = denominator(r(n)) with the rationals r(n) defined above.
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MATHEMATICA
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Denominator[Table[Sum[CatalanNumber[k]/2^(2*(k - 1)), {k, 1, n}], {n, 1, 50}]] (* G. C. Greubel, Feb 08 2017 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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