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Search: id:A100067
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| A100067 |
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Sum binomial(n,k)2^(n-2k), k=0..floor(n/2). |
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+0
4
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| 1, 2, 6, 14, 38, 92, 240, 590, 1510, 3740, 9476, 23564, 59372, 147968, 371636, 927374, 2324870, 5805740, 14538660, 36322340, 90898228, 227153192, 568235696, 1420236524, 3551943388, 8878506392, 22201466280, 55498465400, 138766221800
(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/(1-2x), 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).
Hankel transform is A088138(n+1). [From Paul Barry (pbarry(AT)wit.ie), Jun 16 2009]
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FORMULA
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G.f.: x/((sqrt(1-4x^2)(sqrt(1-4x^2)+x-1); a(n)=sum{k=0..floor(n/2), binomial(n, k)2^(n-2k)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k)2^k/2}.
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CROSSREFS
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Cf. A027306, A100068, A100069.
Sequence in context: A026598 A006864 A071636 this_sequence A026597 A122112 A000634
Adjacent sequences: A100064 A100065 A100066 this_sequence A100068 A100069 A100070
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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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