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A058622
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2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2).
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4
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0, 1, 1, 4, 5, 16, 22, 64, 93, 256, 386, 1024, 1586, 4096, 6476, 16384, 26333, 65536, 106762, 262144, 431910, 1048576, 1744436, 4194304, 7036530, 16777216, 28354132, 67108864, 114159428, 268435456, 459312152, 1073741824, 1846943453
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n) is the number of n-digit binary sequences that have more 1's than 0's. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 16 2009]
Maps to the number of walks that end above 0 on the number line with steps being 1 or -1. [from Benjamin Phillabaum (bphillab(AT)gmail.com),
Mar 6 2011]
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REFERENCES
| A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.1.7)
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FORMULA
| a(n) = 2^(n-1) + ((1+(-1)^n)/4)*binomial(n, n/2); a(n) = sum( binomial(n, i), i=0..(n-1)/2)
G.f.: 2*x/((1-2*x)*(1+2*x+((1+2*x)*(1-2*x))^(1/2))). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 27 2003
E.g.f: (e^(2x)-I_0(2x))/2 where I_n is the Modified Bessel Function
- Benjamin Phillabaum (bphillab(AT)gmail.com), Mar 6 2011
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MATHEMATICA
| Table[Sum[Binomial[n, Floor[n/2 + i]], {i, 1, n}], {n, 0, 32}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 16 2009]
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CROSSREFS
| Cf. A027306.
Sequence in context: A025617 A078581 A092809 * A196021 A064294 A057729
Adjacent sequences: A058619 A058620 A058621 * A058623 A058624 A058625
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KEYWORD
| nonn
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AUTHOR
| Yong Kong (ykong(AT)curagen.com), Dec 29 2000
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