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Search: id:A100069
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| A100069 |
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Sum binomial(n,k)4^(n-2k), k=0..floor(n/2). |
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+0
3
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| 1, 4, 18, 76, 326, 1384, 5892, 25036, 106438, 452344, 1922588, 8170936, 34726940, 147589264, 627256088, 2665837516, 11329815878, 48151714264, 204644809932, 869740430056, 3696396920116, 15709686864304, 66766169526008
(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/(1-4x), 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.: x/((sqrt(1-4x^2)(2sqrt(1-4x^2)+2x-2); a(n)=sum{k=0..floor(n/2), binomial(n, k)4^(n-2k)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k)4^k/2}.
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CROSSREFS
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Cf. A027306, A100067, A100068.
Sequence in context: A108012 A017958 A017959 this_sequence A058870 A112619 A037965
Adjacent sequences: A100066 A100067 A100068 this_sequence A100070 A100071 A100072
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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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