A001453 Catalan numbers - 1.
1, 4, 13, 41, 131, 428, 1429, 4861, 16795, 58785, 208011, 742899, 2674439, 9694844, 35357669, 129644789, 477638699, 1767263189, 6564120419, 24466267019, 91482563639, 343059613649, 1289904147323, 4861946401451, 18367353072151, 69533550916003, 263747951750359
Offset: 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.
Crossrefs
Programs
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Magma
[Catalan(n)-1: n in [2..30]]; // Vincenzo Librandi, May 22 2011
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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 *)
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MuPAD
combinat::dyckWords::count(n)-1 $ n = 2..26; // Zerinvary Lajos, May 08 2008
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PARI
a(n)=(2*n)!/n!/(n+1)!-1 \\ Charles R Greathouse IV, Apr 17 2012
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
Extensions
More terms from James Sellers, Sep 08 2000
Comments