A067294 Third column of triangle A028364.

5, 9, 23, 66, 202, 645, 2123, 7150, 24518, 85306, 300390, 1068484, 3833364, 13855085, 50401395, 184392150, 677998830, 2504191470, 9286661010, 34564913820, 129077071500, 483474711330, 1815928888254, 6837969776076, 25809107768252, 97624796791060, 370016208810668

Offset

0,1

Comments

The first two columns give: A000108 (Catalan) and A005807. The next two columns give: A067295-A067296.

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = A028364(n+2, 2) = C(0)*C(n+2) + C(1)*C(n+1) + C(2)*C(n), with the Catalan numbers C(n) = A000108(n).

a(n) = ((11*n^2+28*n+15)/(2*(2*n+1)*(2*n+3)))*C(n+2).

G.f.: (c2(x)*c(x)-(c2(x)-1)/x)/x^2, with c2(x) = 1+x+2*x^2 and c(x) is the g.f. for Catalan numbers A000108.

a(n) ~ 11 * 2^(2*n+1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 09 2025

Mathematica

With[{c = CatalanNumber}, a[n_] := c[0]*c[n+2] + c[1]*c[n+1] + c[2]*c[n]; Array[a, 25, 0]] (* Amiram Eldar, Oct 09 2025 *)

Crossrefs

First differences are in A071747.

Cf. A000108, A005807, A028364, A067295, A067296.

Sequence in context: A074340 A079993 A163607 * A271543 A272256 A132354

Adjacent sequences: A067291 A067292 A067293 * A067295 A067296 A067297

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 05 2002

Status

approved