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Search: id:A130553
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| A130553 |
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Numerators of partial sums for a series for 2*Pi*sqrt(3)/9. |
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+0
2
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| 1, 7, 6, 169, 1523, 133, 72623, 87149, 823077, 15638477, 46915441, 13834041, 224803169, 6936783521, 5587964507, 4157445593923, 12472336782289, 170187831339, 71785227258967, 153825486983593, 4905323862699739
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OFFSET
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1,2
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COMMENT
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Denominators are given in A130554.
The rational sequence r(n):=2*sum(1/(j*binomial(2*j,j)),j=1..n), n>=1, tends, in the limit n->infinity, to 2*Pi*sqrt(3)/9, which is approximately 1.209199577.
With offset 0 the rationals r(n) coincide with sum(1/((2*j+1)*binomial(2*j,j)),j=0..n), n>=0. See e.g. the Sprugnoli reference. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 17 2008]
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.
C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. 2*Eq.10, p.38. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 17 2008]
R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18. Theorem 3.4, second and fourth identity. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 17 2008]
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LINKS
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W. Lang, Rationals and limit.
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FORMULA
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a(n)=numerator(r(n)), n>=1, with the rationals r(n) defined above and taken in lowest terms.
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EXAMPLE
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Rationals r(n): [1, 7/6, 6/5, 169/140, 1523/1260, 133/110, 72623/60060,...].
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CROSSREFS
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Sequence in context: A070425 A163842 A038272 this_sequence A002394 A105167 A109938
Adjacent sequences: A130550 A130551 A130552 this_sequence A130554 A130555 A130556
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KEYWORD
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nonn,frac,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jul 13 2007
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