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 A058622 a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2). 7
 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; text; internal format)
 OFFSET 0,4 COMMENTS a(n) is the number of n-digit binary sequences that have more 1's than 0's. - Geoffrey Critzer, Jul 16 2009 Maps to the number of walks that end above 0 on the number line with steps being 1 or -1. - Benjamin Phillabaum, Mar 06 2011 Chris Godsil observes that a(n) is the independence number of the (n+1)-folded cube graph; proof is by a Cvetkovic's eigenvalue bound to establish an upper bound and a direct construction of the independent set by looking at vertices at an odd (resp., even) distance from a fixed vertex when n is odd (resp., even). - Stan Wagon, Jan 29 2013 Also the number of subsets of {1,2,...,n} that contain more odd than even numbers.  For example, for n=4, a(4)=5 and the 5 subsets are {1}, {3}, {1,3}, {1,2,3}, {1,3,4}. See A014495 when same number of even and odd numbers. - Enrique Navarrete, Feb 10 2018 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) LINKS Eric Weisstein's World of Mathematics, Folded Cube Graph Eric Weisstein's World of Mathematics, Independence Number FORMULA a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2). a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n, i). G.f.: 2*x/((1-2*x)*(1+2*x+((1+2*x)*(1-2*x))^(1/2))). - Vladeta Jovovic, Apr 27 2003 E.g.f: (e^(2x)-I_0(2x))/2 where I_n is the Modified Bessel Function. - Benjamin Phillabaum, Mar 06 2011 Logarithmic derivative of the g.f. of A210736 is a(n+1). - Michael Somos, Nov 22 2014 EXAMPLE G.f. = x + x^2 + 4*x^3 + 5*x^4 + 16*x^5 + 22*x^6 + 64*x^7 + 93*x^8 + ... MATHEMATICA Table[Sum[Binomial[n, Floor[n/2 + i]], {i, 1, n}], {n, 0, 32}] (* Geoffrey Critzer, Jul 16 2009 *) a[n_] := If[n < 0, 0, (2^n - Boole[EvenQ @ n] Binomial[n, Quotient[n, 2]])/2]; (* Michael Somos, Nov 22 2014 *) a[n_] := If[n < 0, 0, n! SeriesCoefficient[(Exp[2 x] - BesselI[0, 2 x])/2, {x, 0, n}]]; (* Michael Somos, Nov 22 2014 *) Table[2^(n - 1) - (1 + (-1)^n) Binomial[n, n/2]/4, {n, 0, 20}] (* Eric W. Weisstein, Mar 21 2018 *) CoefficientList[Series[2 x/((1 - 2 x) (1 + 2 x + Sqrt[(1 + 2 x) (1 - 2 x)])), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 21 2018 *) PROG (PARI) a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n\2); \\ Michel Marcus, Dec 30 2015 (PARI) x='x+O('x^200); concat(0, Vec(2*x/((1-2*x)*(1+2*x+((1+2*x)*(1-2*x))^(1/2))))) \\ Altug Alkan, Dec 30 2015 CROSSREFS Cf. A027306, A210736. Sequence in context: A092809 A250254 A293344 * A196021 A064294 A284869 Adjacent sequences:  A058619 A058620 A058621 * A058623 A058624 A058625 KEYWORD nonn AUTHOR Yong Kong (ykong(AT)curagen.com), Dec 29 2000 STATUS approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)