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%I #8 Sep 08 2022 08:45:28
%S 1,5,25,5,125,3125,15625,78125,78125,390625,9765625,48828125,
%T 244140625,244140625,78125,6103515625,30517578125,152587890625,
%U 152587890625,762939453125,19073486328125,95367431640625,476837158203125
%N Denominators of partial sums of a series for sqrt(5).
%C Denominators of sums over central binomial coefficients scaled by powers of 5.
%C Numerators are given by A123747.
%C For the rationals r(n) see the W. Lang link under A123747.
%H G. C. Greubel, <a href="/A123748/b123748.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k in lowest terms.
%F r(n) = Sum_{k=0..n} (((2*k-1)!!/((2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.
%e a(3)=5 because r(3)= 1+2/5+6/25+4/25 = 9/5 = A123747(3)/a(3).
%p A123748:=n-> denom(sum(binomial(2*k,k)/5^k, k=0..n)); seq(A123748(n), n=0..25); # _G. C. Greubel_, Aug 10 2019
%t Table[Denominator[Sum[Binomial[2*k, k]/5^k, {k,0,n}]], {n, 0, 25}] (* _G. C. Greubel_, Aug 10 2019 *)
%o (PARI) vector(25, n, n--; denominator(sum(k=0,n, binomial(2*k,k)/5^k))) \\ _G. C. Greubel_, Aug 10 2019
%o (Magma) [Denominator( (&+[Binomial(2*k,k)/5^k: k in [0..n]])): n in [0..25]]; // _G. C. Greubel_, Aug 10 2019
%o (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
%o (GAP) List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k,k)/5^k )) ); # _G. C. Greubel_, Aug 10 2019
%Y Cf. A123747, A123749.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 10 2006
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