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%I #10 Aug 30 2019 03:48:29
%S 1,126,15752,393801,246125639,30765704917,3845713114757,
%T 480714139345054,12017853483626636,7511158427266652362,
%U 938894803408331562046,117361850426041445314536,14670231303255180664525012
%N Numerators of partial sums of Catalan numbers scaled by powers of 1/(5*5^2)=1/125.
%C Denominators are given under A121005.
%C This is the third member (p=2) of the second p-family of partial sums of 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) = F(2*p+1)*sqrt(5)/phi^(2*p+1), 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 on this p-family and the other three ones see the W. Lang link under A120996.
%H W. Lang: <a href="/A121004/a121004.txt">Rationals r(n), limit.</a>
%F a(n)=numerator(r(n)) with r(n) := rII(p=2,n) = sum(C(k)/5^(3*k),k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
%e Rationals r(n): [1, 126/125, 15752/15625, 393801/390625,
%e 246125639/244140625, 30765704917/30517578125,...].
%p The value of the series is lim_{n->infinity}(r(n) := rII(2;n)) = 5*(18 - 11*phi) = 5*sqrt(5)/phi^5 = 1.0081306187560 (maple10, 15 digits).
%Y The second member (p=2) is A120786/A120787.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 16 2006
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