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A121012 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(11^2) = 1/121.

%I A121012 #8 Aug 28 2019 17:56:30
%S A121012 1,120,14522,1757157,212616011,25726537289,282991910191,
%T A121012 34242021133072,4143284557101842,501337431409322440,
%U A121012 667280121205808184436,80740894665902790257970,9769648254574237621422382
%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="/A121012/a121012.txt">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.
%K A121012 nonn,frac,easy
%O A121012 0,2
%A A121012 _Wolfdieter Lang_, Aug 16 2006