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Search: id:A121010
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| A121010 |
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Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320. |
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2
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| 1, 319, 51041, 6533247, 5226597607, 1672511234219, 267601797475073, 342530300768093011, 2192193924915795299, 17537551399326362389569, 2806008223892217982335239, 1795845263291019508694523567
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Denominators are given under A121011.
This is the third member (p=3) 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 link under A120996.
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LINKS
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W. Lang: Rationals r(n), limit.
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FORMULA
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a(n)=numerator(r(n)) with r(n) := rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), with F(6)=8 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
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EXAMPLE
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Rationals r(n): [1, 319/320, 51041/51200, 6533247/6553600,
5226597607/5242880000, 1672511234219/1677721600000,...].
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MAPLE
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The limit lim_{n->infinity} (r(n) := rIII(3; n)) = 8*(-29 + 18*phi) = 8*sqrt(5)/phi^6 = 0.9968943824 (maple10, 10 digits).
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CROSSREFS
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The second member is A121008/A121009.
Sequence in context: A153048 A053020 A064905 this_sequence A110289 A055863 A045813
Adjacent sequences: A121007 A121008 A121009 this_sequence A121011 A121012 A121013
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KEYWORD
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nonn,frac,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 2006
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