%I A121012
%S A121012 1,120,14522,1757157,212616011,25726537289,282991910191,34242021133072,
%T A121012 4143284557101842,501337431409322440,667280121205808184436,
%U A121012 80740894665902790257970,9769648254574237621422382
%N A121012 Numerators of partial alternating sums of Catalan numbers scaled by powers 
               of 1/(11^2) = 1/121.
%C A121012 Denominators are given under A121013.
%C A121012 This is the second member (p=2) of the fourth (normalized) p-family of 
               partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/
               L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) 
               = L(2*p+1)/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 A121012 The partial sums of the above mentioned fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/
               L(2*p+1)^(2*k),k=0..n), n>=0, for p=1,...
%C A121012 For more details on this p-family and the other three ones see the W. 
               Lang link under A120996.
%H A121012 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A121012.text">
               Rationals r(n), limit.</a>
%F A121012 a(n)=numerator(r(n)) with r(n) := rIV(p=2,n) = sum(((-1)^k)*C(k)/L(2*2+1)^(2*k), 
               k=0..n), with L(5)=11 and C(k):=A000108(k) (Catalan). The rationals 
               r(n) are given in lowest terms.
%e A121012 Rationals r(n): [1, 120/121, 14522/14641, 1757157/1771561,
%e A121012 212616011/214358881, 25726537289/25937424601,...].
%p A121012 The limit lim_{n->infinity} (r(n) := rIV(2;n)) = 11*(-8 + 5*phi) = 11/
               phi^5 = 0.9918693812443 (maple10, 10 digits).
%Y A121012 The first member is A120794/A120785. The third member is A121498/A121499.
%Y A121012 Sequence in context: A104592 A135379 A059063 this_sequence A151985 A105188 
               A151604
%Y A121012 Adjacent sequences: A121009 A121010 A121011 this_sequence A121013 A121014 
               A121015
%K A121012 nonn,frac,easy
%O A121012 0,2
%A A121012 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 
               2006