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%I #17 Sep 08 2022 08:45:28
%S 1,3,21,17,99,2223,12039,56763,59337,286961,7358781,36088473,
%T 183146521,181066401,36534213,4535753121,22798981683,113528187171,
%U 113891192583,568042152363,14228623114839,71035463999307,355598139789279
%N Numerators of partial sums of a series for sqrt(5)/3.
%C Denominators are given by A124398.
%C The alternating sums over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/5^k, have the limit s = lim_{n-> infinity} r(n) = sqrt(5)/3. From the expansion of 1/sqrt(1+x) for x=4/5.
%H G. C. Greubel, <a href="/A124397/b124397.txt">Table of n, a(n) for n = 0..1000</a>
%H Wolfdieter Lang, <a href="/A124397/a124397.txt">Rationals and more</a>.
%F a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k in lowest terms.
%F 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.
%e a(3)=17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).
%p seq(numer(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # _G. C. Greubel_, Dec 25 2019
%t Table[Numerator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k,0,n}]], {n,0,20}] (* _G. C. Greubel_, Dec 25 2019 *)
%o (PARI) a(n) = numerator(sum(k=0, n, ((-1)^k)*binomial(2*k,k)/5^k)); \\ _Michel Marcus_, Aug 11 2019
%o (Magma) [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Dec 25 2019
%o (Sage) [numerator(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
%o (GAP) List([0..20], n-> NumeratorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # _G. C. Greubel_, Dec 25 2019
%Y Cf. A123747/A123748 partial sums for a series for sqrt(5).
%Y Cf. A123749/A124396 partial sums for a series for 3/sqrt(5).
%Y Cf. A124398 (denominators), A208899 (sqrt(5)/3).
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 10 2006
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