%I #3 Mar 31 2012 13:20:12
%S 1,320,51200,6553600,5242880000,1677721600000,268435456000000,
%T 343597383680000000,2199023255552000000,17592186044416000000000,
%U 2814749767106560000000000,1801439850948198400000000000
%N Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320.
%C Numerators are given under A121010.
%C This is the third member (p=3) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi) = F(2*p)*sqrt(5)/phi^(2*p), 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 third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=1,...
%C For more details on this p-family and the other three ones see the W. Lang links under A120996 and A121010.
%F a(n)=denominator(r(n)) with r(n) := rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), with F(6)=8 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
%e Rationals r(n): [1, 319/320, 51041/51200, 6533247/6553600,
%e 5226597607/5242880000, 1672511234219/1677721600000,...].
%Y The second member is A121008/A121009.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 16 2006
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