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A045621 a(n) = 2^n - binomial(n, [n/2]). 9
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; text; internal format)
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 n-step 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 1-ascents 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*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 n-th 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..n-1, C(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^(n-2*k)*A000108(k) ). - Paul Barry, Sep 01 2009

a(0)=0, a(n) = 2*a(n-1) if n is even; a(n) = 2*a(n-1) + A000108((n-1)/2) if n is odd. - Vincenzo Librandi, Jun 07 2011

n+1)*a(n) +2*(-n-1)*a(n-1) +4*(-n+2)*a(n-2) +8*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012

MATHEMATICA

Table[2^n - Binomial[n, Floor[n/2]], {n, 0, 30}] (* 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 */

(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

CROSSREFS

Cf. A001405, A000108, A054336, A000346, A068551.

Sequence in context: A094537 A135098 A136488 * A026655 A244398 A100938

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

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Last modified February 18 05:48 EST 2018. Contains 299298 sequences. (Running on oeis4.)