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A123748
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Denominators of partial sums of a series for sqrt(5).
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6
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1, 5, 25, 5, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 78125, 6103515625, 30517578125, 152587890625, 152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125
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OFFSET
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0,2
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COMMENTS
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Denominators of sums over central binomial coefficients scaled by powers of 5.
For the rationals r(n) see the W. Lang link under A123747.
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LINKS
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FORMULA
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a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k in lowest terms.
r(n) = Sum_{k=0..n} (((2*k-1)!!/((2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.
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EXAMPLE
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a(3)=5 because r(3)= 1+2/5+6/25+4/25 = 9/5 = A123747(3)/a(3).
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MAPLE
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MATHEMATICA
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Table[Denominator[Sum[Binomial[2*k, k]/5^k, {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)
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PROG
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(PARI) vector(25, n, n--; denominator(sum(k=0, n, binomial(2*k, k)/5^k))) \\ G. C. Greubel, Aug 10 2019
(Magma) [Denominator( (&+[Binomial(2*k, k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019
(Sage) [denominator( sum(binomial(2*k, k)/5^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Aug 10 2019
(GAP) List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/5^k )) ); # G. C. Greubel, Aug 10 2019
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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