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A045621
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2^n - binomial(n, [n/2]).
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9
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0, 1, 2, 5, 10, 22, 44, 93, 186, 386, 772, 1586, 3172, 6476, 12952, 26333, 52666, 106762, 213524, 431910, 863820, 1744436, 3488872, 7036530, 14073060, 28354132, 56708264, 114159428, 228318856, 459312152, 918624304, 1846943453, 3693886906
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| p(n) = a(n)/2^n is the probability that a majority of heads had occurred at some point after n flips of a fair coin. For example, after 3 flips of a coin, the probability is 5/8 that a majority of heads had occurred at some point. (First flip is heads, p=1/2, or sequence THH, p=1/8.) - Brian L. Galebach (sequence(AT)ProbabilitySports.com), May 14 2001
Hankel transform is (-1)^n*n. - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
Hankel transform of a(n+1) is A127630. [From Paul Barry (pbarry(AT)wit.ie), Sep 01 2009]
a(n) is the number of n-step walks on the number line that are positive at some point along the walk. - Benjamin Phillabaum (bphillab(AT)gmail.com), Mar 6 2011.
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REFERENCES
| S. Mason and J. Parsley, A geometric and combinatorial view of weighted voting, Arxiv preprint arXiv:1109.1082, 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..247
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FORMULA
| a(2*k) = 2*a(2*k-1), a(2*k+1) = 2*a(2*k)+Catalan(k).
a(n+1)=b(0)*b(n)+b(1)*b(n-1)+...+b(n)*b(0), b(k)=C(k, [ k/2 ]).
G.f.: c(x^2)*x/(1-2*x) where c(x) = g.f. for Catalan numbers A000108.
a(n)= A054336(n, 1) (second column of triangle).
E.g.f.: exp(2*x) - I_0(2*x) - I_1(2*x) where I_n(x) is nth modified Bessel function as a function of x. - Benjamin Phillabaum (bphillab(AT)gmail.com), Mar 6 2011.
a(2*n+1)=A000346(n); a(2*n)=A068551(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 16 2003
a(n)=sum(k=0..n-1, C(n, floor(k/2)) ). - Paul Barry (pbarry(AT)wit.ie), Aug 05 2004
a(n+1)=2*a(n)+Catalan(n/2)(1+(-1)^n)/2. - Paul Barry (pbarry(AT)wit.ie), Aug 05 2004
a(n+1)=sum(k=0..floor(n/2), 2^(n-2*k)*A000108(k) ). [From Paul Barry (pbarry(AT)wit.ie), Sep 01 2009]
a(0)=0, a(n) = 2*a(n-1) if n is even; a(n) = 2*a(n-1) + A000180((n-1)/2) if n is odd. [From Vincenzo Librandi, Jun 07 2011]
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MATHEMATICA
| Table[2^n - Binomial[n, Floor[n/2]], {n, 0, 30}] - Roger Bagula (rlbagulatftn(AT)yahoo.com), Aug 26 2006
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PROG
| (PARI) {a(n)=if(n<0, 0, 2^n -binomial(n, n\2))} /* Michael Somos Oct 31 2006 */
(PARI)
x='x+O('x^55); /* That many terms */
egf=exp(2*x)-besseli(0, 2*x)-x*besseli(1, 2*x); /* e.g.f., note Pari omits the overall power of x */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Mar 08 2011 */
(MAGMA) [2^n - Binomial(n, Floor(n/2)): n in [0..32]]; // Bruno Berselli, Mar 08 2011
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CROSSREFS
| a(n) = 2^n-A001405[n].
Sequence in context: A093370 A094537 A135098 * A026655 A100938 A018004
Adjacent sequences: A045618 A045619 A045620 * A045622 A045623 A045624
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KEYWORD
| nonn
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AUTHOR
| David M Bloom, Brooklyn College.
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 08 2006. Adjustments to formulae (correcting offsets) from Michael Somos, Oct 31 2006
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