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A119951
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Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108.
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4
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1, 3, 29, 65, 281, 595, 9949, 20613, 84883, 173965, 1421113, 2894229, 11762641, 23859587, 773201629, 1564082093, 6321150767, 12761711209, 102977321267, 207595672639, 836499257311, 1684433835077, 27122471168057, 54567418372945, 219485160092143, 441266239318305, 3547513302275441
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OFFSET
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1,2
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COMMENTS
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For the corresponding denominator sequence see A120069.
The asymptotics for C(n)/2^(2*(k-1)) is 4/(sqrt(Pi)*k^(3/2)) (see the E. Weisstein link, also for references). 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_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
The above partial sums are equal to 4 - binomial(2n+2,n+1)/2^(2n-1). - Pieter Mostert, Oct 12 2012
The series s = Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers), converges by J. L. Raabe's criterion. See the Meschkowski reference for Raabe's criterion and the example given there. The series he gives as an example can be rewritten as (1 + 4*s)/2. From the expansion of sqrt(1+x) for |x|<=1 one finds for x=-1 the value s=4 (see the W. Lang link).
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REFERENCES
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H. Meschkowski, Unendliche Reihen, 2., verb. u. erw. Aufl., Mannheim, Bibliogr. Inst., 1982, p. 32.
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LINKS
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FORMULA
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a(n) = numerator of Sum_{k=1..n} C(k)/2^(2*(k-1)).
a(n-1) = numerator of (1/4^n)*Sum_{i=0..n-1} (binomial(2*(i+1), i+1)*binomial(2*(n-i), n-i)), for n>=1. - Johannes W. Meijer, May 24 2009
a(n) = (2^n-(2*n+2)!/(2^(n+1)*(n+1)!^2))*gcd((n+1)!,2^(n+1)). - Gary Detlefs, Nov 06 2020
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EXAMPLE
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Rationals r(n): [1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, ...]
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MATHEMATICA
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Numerator[Table[(1/4^n)*Sum[Binomial[2*(i + 1), i + 1]*Binomial[2*(n - i), n - i], {i, 0, n - 1}], {n, 1, 50}]] (* G. C. Greubel, Jan 31 2017 *)
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PROG
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(PARI) for(n=1, 25, print1(numerator(sum(i=0, n-1, binomial(2*(i+1), i+1)* binomial(2*(n-i), n-i))/4^n), ", ")) \\ G. C. Greubel, Jan 31 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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STATUS
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approved
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