A077587 a(n) = C(n+1) + n*C(n) where C = A000108 (Catalan numbers).

1, 3, 9, 29, 98, 342, 1221, 4433, 16302, 60554, 226746, 854658, 3239044, 12332140, 47137005, 180780345, 695367510, 2681600130, 10364759790, 40142121030, 155748675420, 605274171060, 2355676013730, 9180275261274, 35819645937228

Offset: 0

Author

Michael Somos, Nov 09 2002

Keywords

• nonn
• easy

Comments

Number of ascents of length 2 starting at an even level in all Dyck paths of semilength n+2. Example: a(1)=3 because all Dyck paths of semilength 3 are UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and UUUDDD, where U=(1,1), D=(1,-1), having altogether 3 ascents of length 2 that start at an even level (shown between parentheses). - Emeric Deutsch, Nov 29 2005

a(n) is the number of parking functions of size n+1 avoiding the patterns 132, 231, and 321. - Lara Pudwell, Apr 10 2023

Links

• G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 0..200 from Vincenzo Librandi)
• Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
• A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv:1502.04925 [cs.CG], 2015.

Crossrefs

Cf. A000108, A057977, A114462.

Programs

Maple

egf := x -> exp(2*x)*(1+1/x)*BesselI(1, 2*x);
seq(n!*coeff(series(egf(x), x, n+2), x, n), n=0..24); # Peter Luschny, Apr 14 2014

Mathematica

Table[(CatalanNumber[n + 1] + n CatalanNumber[n]), {n, 0, 40}] (* Vincenzo Librandi, Apr 15 2014 *)

PARI

a(n)=if(n<0,0,(n^2+6*n+2)*(2*n)!/n!/(n+2)!)

PARI

a(n)=if(n<0,0,polcoeff((4+x+1/x-(x+1/x)^2)*(1+x)^(2*n),n)/2)

Formula

a(n) = binomial(2n+1, n+1) - binomial(2n, n+2).

a(n) = (3*(3*n+2)*a(n-1) - 2*(11*n-7)*a(n-2) + 4*(2*n-5)*a(n-3))/(n+2), n>2.

G.f.: A(x) = (1 - 3*x - (1-5*x+2*x^2)/sqrt(1-4*x) )/(2*x^2) satisfies 0 = (x^2+4*x-1) + (12*x^2-7*x+1)*A + (4*x^3-x^2)*A^2.

E.g.f.: A(x) = (1+x)B(x)' where B(x) = e.g.f. of A000108.

a(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*2^(n-k); here the A057977 are understood as the extended Catalan numbers (see also A063549). Related to Touchard's identity. - Peter Luschny, Jul 14 2016

a(n) ~ 4^n/sqrt(Pi*n). - Ilya Gutkovskiy, Jul 14 2016

Asymptotic starts a(n) ~ (4^n/sqrt(Pi*n))*(1 + (23/2^3)/n - (1199/2^7)/n^2 +(22685/2^10)/n^3 - (1562421/2^15)/n^4 + ... ). - Peter Luschny, Jul 14 2016