OFFSET
2,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 2..500 (first 170 terms from Vincenzo Librandi)
R. M. Baer and P. Brock, Natural sorting over permutation spaces, Math. Comp. 22 1968 385-410.
J. M. Hammersley, A few seedings of research, in Proc. Sixth Berkeley Sympos. Math. Stat. and Prob., ed. L. M. le Cam et al., Univ. Calif. Press, 1972, Vol. I, pp. 345-394.
Piera Manara and Claudio Perelli Cippo, The fine structure of 4321 avoiding involutions and 321 avoiding involutions, PU. M. A. Vol. 22 (2011), 227-238.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
FORMULA
a(n) = A000108(n) - 1 = binomial(2*n,n)/(n+1) - 1.
D-finite with recurrence: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(9*n-13)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Sep 04 2013
a(n) = Sum_{k=1..floor(n/2)} (C(n,k)-C(n,k-1))^2. - J. M. Bergot, Sep 17 2013
a(n) = Sum_{k=1..n-1} A000245(n-k-1). - John M. Campbell, Dec 28 2016
From Ilya Gutkovskiy, Dec 28 2016: (Start)
O.g.f.: (1 - sqrt(1 - 4*x))/(2*x) - 1/(1 - x).
E.g.f.: exp(x)*(exp(x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1). (End)
a(n)= 3*Sum_{k=1..n} binomial(2*k-2,k)/(k+1). - Gary Detlefs, Feb 14 2020
MAPLE
with(combstruct): bin := {B=Union(Z, Prod(B, B))}: seq(count([B, bin, unlabeled], size=n+1)-1, n=2..30); # Zerinvary Lajos, Dec 05 2007
MATHEMATICA
Array[CatalanNumber, 30, 2] - 1 (* Jean-François Alcover, Mar 11 2014 *)
PROG
(MuPAD) combinat::dyckWords::count(n)-1 $ n = 2..26; // Zerinvary Lajos, May 08 2008
(Magma) [Catalan(n)-1: n in [2..30]]; // Vincenzo Librandi, May 22 2011
(PARI) a(n)=(2*n)!/n!/(n+1)!-1 \\ Charles R Greathouse IV, Apr 17 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Sep 08 2000
STATUS
approved