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Search: id:A121008
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| A121008 |
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Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*3^2) = 1/45. |
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+0
6
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| 1, 44, 1982, 17837, 4013339, 60200071, 2709003239, 121905145612, 658287786362, 740573759652388, 33325819184374256, 1499661863296782734, 67484783848355431042, 607363054635198730798, 3036815273175993713422
(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 A121009.
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).
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=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.
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EXAMPLE
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Rationals r(n): [1, 44/45, 1982/2025, 17837/18225, 4013339/4100625,
60200071/61509375, 2709003239/2767921875,...].
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MAPLE
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The limit lim_{n->infinity}(r(n) := rIII(2; n)) = 3*(-11 + 7*phi) = 3*sqrt(5)/phi^4 = 0.9787137637479 (maple10, 15 digits).
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CROSSREFS
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The first member (p=1) is A121006/A121007.
Sequence in context: A009988 A041925 A094506 this_sequence A078279 A007105 A119096
Adjacent sequences: A121005 A121006 A121007 this_sequence A121009 A121010 A121011
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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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