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Search: id:A100068
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| A100068 |
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Sum binomial(n,k)3^(n-2k), k=0..floor(n/2). |
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+0
3
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| 1, 3, 11, 36, 123, 408, 1370, 4560, 15235, 50760, 169326, 564336, 1881582, 6271632, 20907156, 69689376, 232304355, 774343560, 2581169510, 8603882160, 28679699578, 95598937008, 318663476076, 1062211351776, 3540705857998
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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An inverse Chebyshev transform of x/(1-3x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4x^2))A(xc(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, sum{k=0..floor(n/2), binomial(n,k)r^(n-k)} has g.f. 2x/((sqrt(1-4x^2)(r*sqrt(1-4x^2)+r*x-r).
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FORMULA
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G.f.: 2x/((sqrt(1-4x^2)(3sqrt(1-4x^2)+2x-3); a(n)=sum{k=0..floor(n/2), binomial(n, k)3^(n-k)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k)3^k/2}.
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CROSSREFS
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Cf. A027306, A100067, A100069.
Sequence in context: A119088 A017937 A017938 this_sequence A119213 A068644 A119054
Adjacent sequences: A100065 A100066 A100067 this_sequence A100069 A100070 A100071
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Nov 02 2004
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