0,2

Proof by induction.

a(n) = total area below paths consisting of steps east (1,0) and north (0,1) from (0,0) to (n+2,n+2) that stay weakly below y=x. For example, the two paths with n=0 are

. _|.....|

_|.....__|

The first has area 1 below it, the second area 0, and so a(0)=1. - David Callan (callan(AT)stat.wisc.edu), Dec 09 2004

a(n) = 4^(n+1)*sum(binomial(2k, k-1)/4^k, k=1..n+1) = ((n+3)^2)*C(n+2)/2-2^(2*n+3), C = Catalan. Also a(n+1)=4*a(n)+binomial(2(n+2), n+1).

G.f.: diff(c(x), x)/(1-4*x), where c(x) = g.f. for Catalan numbers; convolution of A001791 and powers of 4. G.f. also c(x)^2/(1-4*x)^(3/2); convolution of Catalan numbers A000108 C(n), n >= 1, with A002457; convolution of A008549(n), n >= 1, with A000984 (central binomial coefficients).

a(n) = Sum_{k, 0<=k<=n+1}A039598(n+1,k)*k^2. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 16 2007

Cf. A000108, A002457, A008549, A000984.

Sequence in context: A099110 A106393 A139262 this_sequence A026900 A016198 A051140

Adjacent sequences: A029757 A029758 A029759 this_sequence A029761 A029762 A029763

nonn

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)