A123749 - OEIS

%I #17 Sep 08 2022 08:45:28

%S 1,11,35,965,8755,8783,237449,2138185,6415985,519743405,4677875401,

%T 14033861347,378916960525,3410263045325,3410267502725,30692424759805,

%U 276231889624955,828695755304725,67124359204727825,604119244624305025

%N Numerators of partial sums of a series for 3/sqrt(5) = (3/5)*sqrt(5).

%C Denominators are given by A124396.

%C The sums over central binomial coefficients scaled by powers of 9, r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k have the limit s = lim_{n->infinity} r(n) = 3/sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/9.

%H G. C. Greubel, <a href="/A123749/b123749.txt">Table of n, a(n) for n = 0..1000</a>

%H Wolfdieter Lang, <a href="/A123749/a123749.txt">Rationals and more.</a>

%F a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k in lowest terms.

%F r(n) = Sum_{k=0..n} (((2*k-1)!!/((2*k)!!)*(4/9)^k, n>=0, with the double factorials A001147 and A000165.

%e a(3) = 965 because r(3) = 1 + 2/9 + 2/27 + 20/729 = 965/729 = a(3)/A124396(3).

%p A123749:=n-> numer(sum(binomial(2*k,k)/9^k, k=0..n)); seq(A123749(n), n=0..20); # _G. C. Greubel_, Aug 10 2019

%t Table[Numerator[Sum[Binomial[2*k, k]/9^k, {k,0,n}]], {n, 0, 20}] (* _G. C. Greubel_, Aug 10 2019 *)

%o (PARI) vector(20, n, n--; numerator(sum(k=0,n, binomial(2*k,k)/9^k))) \\ _G. C. Greubel_, Aug 10 2019

%o (Magma) [Numerator( (&+[Binomial(2*k,k)/9^k: k in [0..n]])): n in [0..20]]; // _G. C. Greubel_, Aug 10 2019

%o (Sage) [numerator( sum(binomial(2*k,k)/9^k for k in (0..n)) ) for n in (0..20)] # _G. C. Greubel_, Aug 10 2019

%o (GAP) List([0..20], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/9^k )) ); # _G. C. Greubel_, Aug 10 2019

%Y Cf. A124396 (denominators).

%Y Cf. A123747/A123748 partial sums for a series for sqrt(5).

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 10 2006