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A124398
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Denominators of partial sums of a series for sqrt(5)/3.
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2
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1, 5, 25, 25, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 48828125, 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 alternating sums over central binomial coefficients scaled by powers of 5.
For the rationals r(n) see the W. Lang link under A124397.
r(n) is not 1/3 times the rational sequence A123747/A123748 which converges to sqrt(5).
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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} (-1)^k * binomial(2*k,k)/5^k, in lowest terms.
r(n) = Sum_{k=0..n} (-1)^k*((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) = 25 because r(3)= 1 - 2/5 + 6/25 - 4/25 = 17/25 = A124397(3)/a(3).
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MAPLE
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seq(denom(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
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MATHEMATICA
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Table[Denominator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Dec 25 2019 *)
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PROG
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(PARI) a(n) = denominator(sum(k=0, n, ((-1)^k)*binomial(2*k, k)/5^k)); \\ Michel Marcus, Aug 11 2019
(Magma) [Denominator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
(Sage) [denominator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019
(GAP) List([0..20], n-> DenominatorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 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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