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%I #8 Aug 28 2019 17:31:06
%S 1,44,1982,17837,4013339,60200071,2709003239,121905145612,
%T 658287786362,740573759652388,33325819184374256,1499661863296782734,
%U 67484783848355431042,607363054635198730798,3036815273175993713422
%N Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*3^2) = 1/45.
%C Denominators are given under A121009.
%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.
%H W. Lang: <a href="/A121008/a121008.txt">Rationals r(n), limit.</a>
%F a(n)=numerator(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,...].
%p The limit lim_{n->infinity}(r(n) := rIII(2;n)) = 3*(-11 + 7*phi) = 3*sqrt(5)/phi^4 = 0.9787137637479 (maple10, 15 digits).
%Y The first member (p=1) is A121006/A121007.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 16 2006
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