A037955 a(n) = binomial(n, floor(n/2)-1).

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, 565722720, 1037158320, 2203961430, 4059928950, 8597496600, 15905368710, 33578000610, 62359143990

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

a(n) is the number of subsets of {1,2,...,n} that contain exactly 1 more even than odd elements. For example, a(6) = 15 and the 15 sets are {2}, {4}, {6}, {1,2,4}, {1,2,6}, {1,4,6}, {2,3,4}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {4,5,6}, {1,2,3,4,6}, {1,2,4,5,6}, {2,3,4,5,6}. - Enrique Navarrete, Dec 20 2019

Links

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

Crossrefs

Cf. A037952, A047182.

Programs

Magma

[Binomial(n, Floor((n-2)/2)): n in [0..40]]; // G. C. Greubel, Dec 31 2019

Maple

seq(binomial(n, floor((n-2)/2)), n = 0..40);

Mathematica

Table[Binomial[n,Floor[n/2-1]], {n,0,40}] (* Wesley Ivan Hurt, Oct 16 2013 *)

PARI

vector(41, n, binomial(n-1, (n-3)\2) ) \\ G. C. Greubel, Dec 31 2019

Sage

[binomial(n, floor(n/2)-1) for n in (0..40)] # G. C. Greubel, Dec 31 2019

Formula

E.g.f.: Bessel_I(2,2*x) + Bessel_I(3,2*x). - 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

a(n) = binomial(n, (n-2)/2), n even; a(n) = binomial(n, (n-3)/2), n odd. - Enrique Navarrete, Dec 20 2019