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A037955 a(n) = C(n, floor(n/2)-1). 10
0, 0, 1, 1, 4, 5, 15, 21, 56, 84, 210, 330, 792, 1287, 3003, 5005, 11440, 19448, 43758, 75582, 167960, 293930, 646646, 1144066, 2496144, 4457400, 9657700, 17383860, 37442160, 67863915, 145422675, 265182525 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Number of returns to the axis in all left factors of Dyck paths of length n. Example: a(4)=4 because in U(D)U(D), U(D)UU, UUD(D), UUDU, UUUD, and UUUU we have a total of 2+1+1+0+0+0=4 returns to the axis (shown between parentheses); here U=(1,1) and D=(1,-1). - Emeric Deutsch, Jun 06 2011

LINKS

Robert Israel, Table of n, a(n) for n = 0..3324

FORMULA

E.g.f.: Bessel_I(2,2x)+Bessel_I(3,2x). - Paul Barry, Feb 28 2006

G.f.: g(z) = z^2*c^3/(1-z*c), where c = (1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2. - Emeric Deutsch, Jun 06 2011

(n+3)*(n-2)*a(n) +2*n*a(n-1) +4*n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 30 2012

MAPLE

seq(binomial(n, floor((n-2)*1/2)), n = 0 .. 37);

MATHEMATICA

Table[Binomial[n, Floor[n/2-1]], {n, 0, 50}] (* Wesley Ivan Hurt, Oct 16 2013 *)

CROSSREFS

Cf. A037952, A047182.

Sequence in context: A230983 A100234 A007390 * A225121 A267991 A225536

Adjacent sequences:  A037952 A037953 A037954 * A037956 A037957 A037958

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)