A026008 a(n) = T(n, floor(n/2)), where T = Catalan triangle (A008315).

1, 1, 2, 3, 5, 9, 14, 28, 42, 90, 132, 297, 429, 1001, 1430, 3432, 4862, 11934, 16796, 41990, 58786, 149226, 208012, 534888, 742900, 1931540, 2674440, 7020405, 9694845, 25662825, 35357670, 94287120, 129644790, 347993910, 477638700, 1289624490, 1767263190

Offset

0,3

Comments

a(n) is the number of Catalan paths in Quadrant I from (0,0) to (n, gcd(n,2)). - Clark Kimberling, Jun 26 2004

Links

Table of n, a(n) for n=0..36.

Formula

a(2n) = A000108(n+1), a(2n+1) = A000245(n+1).

a(2n) = C(2n+2, n+1)/(n+2), a(2n+1) = 3C(2n+2, n)/(n+3). - Ralf Stephan, Apr 30 2004

Conjecture: (n+5)*a(n) + (n+3)*a(n-1) + (-5*n-9)*a(n-2) - 4*n*a(n-3) +4*(n-2)*a(n-4) = 0. - R. J. Mathar, Jun 10 2013

a(n) ~ c * 2^(n+5/2) / (n^(3/2) * sqrt(Pi)), where c = 3 is n is odd, and c = 2 if n is even. - Amiram Eldar, Sep 24 2025

Mathematica

a[n_] := If[EvenQ[n], Binomial[n+2, n/2+1]/(n/2 + 2), 3*Binomial[n+1, (n-1)/2]/((n +5 )/2)]; Array[a, 40, 0] (* Amiram Eldar, Sep 24 2025 *)

Crossrefs

Cf. A000108, A000245, A008315.

Sequence in context: A032089 A352079 A105044 * A101461 A384337 A085897

Adjacent sequences: A026005 A026006 A026007 * A026009 A026010 A026011

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved