0,3

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(2k) = 2a(2k-1), a(2k+1) = 2a(2k)+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).

a(2n+1)=A000346(n); a(2n)=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)=2a(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-2k)*A000108(k)}. [From Paul Barry (pbarry(AT)wit.ie), Sep 01 2009]

Table[2^n - Binomial[n, Floor[n/2]], {n, 0, 30}] - Roger Bagula (rlbagulatftn(AT)yahoo.com), Aug 26 2006

(PARI) {a(n)=if(n<0, 0, 2^n -binomial(n, n\2))} /* Michael Somos Oct 31 2006 */

a(n) = 2^n-A001405[n].

Sequence in context: A093370 A094537 A135098 this_sequence A026655 A100938 A018004

Adjacent sequences: A045618 A045619 A045620 this_sequence A045622 A045623 A045624

nonn

David M Bloom, Brooklyn College.

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