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URL: https://oeis.org/A026008

⇱ A026008 - OEIS

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A026008
a(n) = T(n, floor(n/2)), where T = Catalan triangle (A008315).
3
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
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
KEYWORD
nonn,easy
STATUS
approved