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%I #7 Aug 29 2019 16:24:12
%S 1,21,211,1689,84457,1689161,16891643,1351331869,2702663881,
%T 270266390531,2702663909509,108106556409753,1081065564149533,
%U 4324262256635277,43242622566419631,6918819610629079929
%N Numerators of partial sums of Catalan numbers scaled by powers of 1/20.
%C Denominators are given under A120787.
%C From the expansion of 2*sqrt(5)/5 = sqrt(1-1/5) = 1-(1/10)*sum(C(k)/20^k,k=0..infinity) one has r:=limit(r(n),n to infinity)= 2*(5 - 2*sqrt(5)) = 2*(7 - 4*phi) = 1.055728090..., where phi:= (1+sqrt(5))/2 (golden section) and the partial sums r(n) are defined below.
%C This is the second member (p=1) in the second p-family of partial sums of the normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
%C The partial sums of the above mentioned second p-family are rII(p;n):=sum(C(k)/((5^k)*F(2*p+1)^(2*k),k=0..n), n>=0, for p=0,1,...
%C For more details about this p-family and the other three ones see the W. Lang link under A120996.
%H W. Lang: <a href="/A120786/a120786.txt">Rationals r(n) and limit.</a>
%F a(n)=numerator(r(n)), with the rationals r(n):=sum(C(k)/20^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
%e Rationals r(n): [1, 21/20, 211/200, 1689/1600, 84457/80000,
%e 1689161/1600000, 16891643/16000000, 1351331869/1280000000,...].
%K nonn,easy,frac
%O 0,2
%A _Wolfdieter Lang_, Jul 20 2006
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