0,2

Second binomial transform of A032443 with interpolated zeros.

a(n) = total number of lattice points, taken over all Dyck n-paths (A000108), that (i) lie on or above ground level and (ii) lie on or directly below a peak. For example with n=2, UUDD has 1 peak contributing 3 lattice points--(2,0), (2,1) and (2,2) when the path starts at the origin--and UDUD has 2 peaks, each contributing 2 lattice points and so a(2)=3+4=7. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006

Hankel transform is binomial(n+2,2). - Paul Barry (pbarry(AT)wit.ie), Dec 04 2007

Image of (-1)^n under the Riordan array ((1/2)(1/(1-4x)+1/sqrt(1-4x)),c(x)-1), c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), Jun 15 2008

a(n)=sum{k=0..n, C(n,k)*2^(n-k-2)*(2^k+C(k,k/2))(1+(-1)^k)}; a(n)=(A000984(n)+A081294(n))/2.

G.f.: (1-4x+(1-2x)sqrt(1-4x))/(2(1-4x)^(3/2)); a(n)=sum{k=0..n, sum{j=0..n, C(2n,n-k-j)(-1)^j}}; - Paul Barry (pbarry(AT)wit.ie), Jun 15 2008

a(n)=sum{k=0..n, C(2n,n-k)*(1+(-1)^k)/2}. [From Paul Barry (pbarry(AT)wit.ie), Aug 06 2009]

Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 07 2009: (Start)

a(n)=C(2n-1,n-1)+(4^n+3*0^n)/4.

Integral representation a(n)=(1/(2*pi))*Int(x^n/sqrt(x(4-x)),x,0,4)+(4^n+0^n)/4. (End)

Adjacent sequences: A114118 A114119 A114120 this_sequence A114122 A114123 A114124

Sequence in context: A001075 A113436 A126223 this_sequence A049775 A101850 A045868

easy,nonn

Paul Barry (pbarry(AT)wit.ie), Feb 13 2006