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A121007 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/5. 3


%S 1,5,25,25,625,3125,15625,78125,78125,1953125,9765625,48828125,

%T 244140625,244140625,1220703125,6103515625,30517578125,152587890625,

%U 30517578125,3814697265625,19073486328125

%N Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/5.

%C Numerators are given under A121006.

%C This is the first member (p=1) 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 A121006.

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

%e Rationals r(n): 1, 4/5, 22/25, 21/25, 539/625, 2653/3125,

%e 13397/15625, 66556/78125, 66842/78125, 1666188/1953125, 8347736/9765625,...]

%p The limit lim_{n->infinity} (r(n) := rIII(1;n)) = -4 + 3*phi = 0.85410196624968 (maple10, 15 digits).

%Y The second member (p=2) is A121008/A121009.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 16 2006

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Last modified June 7 01:26 EDT 2020. Contains 334836 sequences. (Running on oeis4.)