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A037955 a(n) = binomial(n, floor(n/2)-1). 11
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 (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
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
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
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 *)
PROG
(PARI) vector(41, n, binomial(n-1, (n-3)\2) ) \\ G. C. Greubel, Dec 31 2019
(Magma) [Binomial(n, Floor((n-2)/2)): n in [0..40]]; // G. C. Greubel, Dec 31 2019
(Sage) [binomial(n, floor(n/2)-1) for n in (0..40)] # G. C. Greubel, Dec 31 2019
CROSSREFS
Sequence in context: A230983 A100234 A007390 * A225121 A267991 A225536
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified February 28 01:36 EST 2024. Contains 370379 sequences. (Running on oeis4.)