A121009 - OEIS

%I #2 Mar 31 2012 13:20:12

%S 1,45,2025,18225,4100625,61509375,2767921875,124556484375,

%T 672605015625,756680642578125,34050628916015625,1532278301220703125,

%U 68952523554931640625,620572711994384765625,3102863559971923828125

%N Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*3^2) = 1/45.

%C Numerators are given under A121008.

%C This is the second member (p=2) 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 link under A120996 and A121008.

%F a(n)=denominator(r(n)) with r(n) := rIII(p=2,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*2)^(2*k)),k=0..n), with F(4)=3 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

%e Rationals r(n): [1, 44/45, 1982/2025, 17837/18225, 4013339/4100625,

%e 60200071/61509375, 2709003239/2767921875,...].

%Y The first member (p=1) is A121006/A121007.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 16 2006