

A045621


a(n) = 2^n  binomial(n, floor(n/2)).


12



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
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OFFSET

0,3


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 Galebach, May 14 2001
Hankel transform is (1)^n*n.  Paul Barry, Jan 11 2007
Hankel transform of a(n+1) is A127630.  Paul Barry, Sep 01 2009
a(n) is the number of nstep walks on the number line that are positive at some point along the walk.  Benjamin Phillabaum, Mar 06 2011


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..247
Kairi Kangro, Mozhgan Pourmoradnasseri, Dirk Oliver Theis, Short note on the number of 1ascents in dispersed dyck paths, arXiv:1603.01422 [math.CO], 2016.
S. Mason and J. Parsley, A geometric and combinatorial view of weighted voting, arXiv preprint arXiv:1109.1082 [math.CO], 2011.


FORMULA

a(n) = 2^n  A001405(n).
a(2*k) = 2*a(2*k1), a(2*k+1) = 2*a(2*k) + Catalan(k).
a(n+1) = b(0)*b(n)+b(1)*b(n1)+...+b(n)*b(0), b(k)=C(k, [ k/2 ]).
G.f.: c(x^2)*x/(12*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, Mar 06 2011
a(2*n+1) = A000346(n); a(2*n) = A068551(n).  Emeric Deutsch, Nov 16 2003
a(n) = Sum_{k=0..n1} binomial(n, floor(k/2)).  Paul Barry, Aug 05 2004
a(n+1) = 2*a(n) + Catalan(n/2)*(1+(1)^n)/2.  Paul Barry, Aug 05 2004
a(n+1) = Sum_{k=0..floor(n/2)} 2^(n2*k)*A000108(k).  Paul Barry, Sep 01 2009
(n+1)*a(n) +2*(n1)*a(n1) +4*(n+2)*a(n2) +8*(n2)*a(n3) = 0.  R. J. Mathar, Dec 02 2012


MAPLE

seq( 2^n binomial(n, floor(n/2)), n=0..35); # G. C. Greubel, Jan 13 2020


MATHEMATICA

Table[2^n  Binomial[n, Floor[n/2]], {n, 0, 35}] (* Roger L. Bagula, Aug 26 2006 *)


PROG

(PARI) {a(n)=if(n<0, 0, 2^n binomial(n, n\2))} /* Michael Somos, Oct 31 2006 */
(MAGMA) [2^n  Binomial(n, Floor(n/2)): n in [0..35]]; // Bruno Berselli, Mar 08 2011
(Sage) [2^n binomial(n, floor(n/2)) for n in (0..35)] # G. C. Greubel, Jan 13 2020
(GAP) List([0..35], n> 2^n  Binomial(n, Int(n/2)) ); # G. C. Greubel, Jan 13 2020


CROSSREFS

Cf. A000108, A001405, A000346, A054336, A068551.
Sequence in context: A094537 A135098 A136488 * A026655 A336484 A244398
Adjacent sequences: A045618 A045619 A045620 * A045622 A045623 A045624


KEYWORD

nonn


AUTHOR

David M Bloom, Brooklyn College


EXTENSIONS

Edited by N. J. A. Sloane, Oct 08 2006
Adjustments to formulas (correcting offsets) from Michael Somos, Oct 31 2006


STATUS

approved



