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A001453
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Catalan numbers - 1.
(Formerly M3464 N1409)
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36
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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
(list;
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internal format)
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OFFSET
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2,2
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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
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MAPLE
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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
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MATHEMATICA
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PROG
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(MuPAD) combinat::dyckWords::count(n)-1 $ n = 2..26; // Zerinvary Lajos, May 08 2008
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CROSSREFS
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A141364 is essentially the same sequence.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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