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%I M3464 N1409 #98 Oct 27 2023 19:12:30
%S 1,4,13,41,131,428,1429,4861,16795,58785,208011,742899,2674439,
%T 9694844,35357669,129644789,477638699,1767263189,6564120419,
%U 24466267019,91482563639,343059613649,1289904147323,4861946401451,18367353072151,69533550916003,263747951750359
%N Catalan numbers - 1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A001453/b001453.txt">Table of n, a(n) for n = 2..500</a> (first 170 terms from Vincenzo Librandi)
%H R. M. Baer and P. Brock, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0228216-8">Natural sorting over permutation spaces</a>, Math. Comp. 22 1968 385-410.
%H J. M. Hammersley, <a href="http://projecteuclid.org/euclid.bsmsp/1200514101">A few seedings of research</a>, 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.
%H Piera Manara and Claudio Perelli Cippo, <a href="http://www.mat.unisi.it/newsito/puma/public_html/22_2/manara_perelli-cippo.pdf">The fine structure of 4321 avoiding involutions and 321 avoiding involutions</a>, PU. M. A. Vol. 22 (2011), 227-238.
%H Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016.
%F a(n) = A000108(n) - 1 = binomial(2*n,n)/(n+1) - 1.
%F 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
%F a(n) = Sum_{k=1..floor(n/2)} (C(n,k)-C(n,k-1))^2. - _J. M. Bergot_, Sep 17 2013
%F a(n) = Sum_{k=1..n-1} A000245(n-k-1). - _John M. Campbell_, Dec 28 2016
%F From _Ilya Gutkovskiy_, Dec 28 2016: (Start)
%F O.g.f.: (1 - sqrt(1 - 4*x))/(2*x) - 1/(1 - x).
%F E.g.f.: exp(x)*(exp(x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1). (End)
%F a(n)= 3*Sum_{k=1..n} binomial(2*k-2,k)/(k+1). - _Gary Detlefs_, Feb 14 2020
%p 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
%t Array[CatalanNumber, 30, 2] - 1 (* _Jean-François Alcover_, Mar 11 2014 *)
%o (MuPAD) combinat::dyckWords::count(n)-1 $ n = 2..26; // _Zerinvary Lajos_, May 08 2008
%o (Magma) [Catalan(n)-1: n in [2..30]]; // _Vincenzo Librandi_, May 22 2011
%o (PARI) a(n)=(2*n)!/n!/(n+1)!-1 \\ _Charles R Greathouse IV_, Apr 17 2012
%Y Cf. A000108, A001454. Column k=2 of A047874.
%Y A141364 is essentially the same sequence.
%Y All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.
%K nonn,easy
%O 2,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Sep 08 2000
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