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A121007 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/5. 3
1, 5, 25, 25, 625, 3125, 15625, 78125, 78125, 1953125, 9765625, 48828125, 244140625, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 30517578125, 3814697265625, 19073486328125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Numerators are given under A121006.

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).

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,...

For more details on this p-family and the other three ones see the W. Lang links under A120996 and A121006.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

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.

EXAMPLE

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

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

MAPLE

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

CROSSREFS

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

Sequence in context: A039936 A124398 A121003 * A043057 A137112 A037410

Adjacent sequences:  A121004 A121005 A121006 * A121008 A121009 A121010

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Aug 16 2006

STATUS

approved

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Last modified April 9 04:05 EDT 2020. Contains 333343 sequences. (Running on oeis4.)