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%I #115 Jul 16 2024 17:03:02
%S 1,1,3,13,67,381,2307,14589,95235,636925,4341763,30056445,210731011,
%T 1493303293,10678370307,76957679613,558403682307,4075996839933,
%U 29909606989827,220510631755773,1632599134961667,12133359132082173
%N Generalized Catalan numbers C(2; n).
%C a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).
%C a(n) = number of Dyck n-paths (A000108) in which each upstep (U) not at ground level is colored red (R) or blue (B). For example, a(3)=3 counts URDD, UBDD, UDUD (D=downstep). - _David Callan_, Mar 30 2007
%C The Hankel transform of this sequence is A002416. - _Philippe Deléham_, Nov 19 2007
%C The sequence a(n)/2^n, with g.f. 1/(1-xc(x)/2), has Hankel transform 1/2^n. - _Paul Barry_, Apr 14 2008
%C The REVERT transform of the odd numbers [1,3,5,7,9,...] is [1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, ...] - _N. J. A. Sloane_, May 26 2017
%H Vincenzo Librandi, <a href="/A064062/b064062.txt">Table of n, a(n) for n = 0..200</a>
%H J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.
%H Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, <a href="https://arxiv.org/abs/2309.00426">A lattice on Dyck paths close to the Tamari lattice</a>, arXiv:2309.00426 [math.CO], 2023.
%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
%H J. Bloom and S. Elizalde, <a href="http://arxiv.org/abs/1211.3442">Pattern avoidance in matchings and partitions</a>, arXiv:1211.3442 [math.CO] (2012) Theorem 6.1.
%H N. Bonichon, C. Gavoille and N. Hanusse, <a href="http://dept-info.labri.fr/~gavoille/article/BGH05">Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation</a>, In Proceedings of WG'03, volume 2880 of LNCS, pp. 81-92, 2003.
%H Alexander Burstein, Sergi Elizalde and Toufik Mansour, <a href="https://arxiv.org/abs/math/0610234">Restricted Dumont permutations, Dyck paths and noncrossing partitions</a>, arXiv:math/0610234 [math.CO], 2006.
%H Xiang-Ke Chang, X.-B. Hu, H. Lei and Y.-N. Yeh, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p8">Combinatorial proofs of addition formulas</a>, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%H Simone Fioravanti, Steve Hanneke, Shay Moran, Hilla Schefler, and Iska Tsubari, <a href="https://arxiv.org/abs/2407.07765">Ramsey Theorems for Trees and a General 'Private Learning Implies Online Learning' Theorem</a>, arXiv:2407.07765 [cs.LG], 2024. See p. 44.
%H Ivan Geffner and Marc Noy, <a href="https://doi.org/10.37236/6249">Counting Outerplanar Maps</a>, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A. Vieru, <a href="http://arxiv.org/abs/1107.2938">Agoh's conjecture: its proof, its generalizations, its analogues</a>, arXiv preprint arXiv:1107.2938 [math.NT], 2011-2012.
%F G.f.: (1 + 2*x*C(2*x)) / (1+x) = 1/(1 - x*C(2*x)) with C(x) g.f. of Catalan numbers A000108.
%F a(n) = A062992(n-1) = Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(2^m)/n, n >= 1, a(0) = 1.
%F a(n) = Sum_{k = 0..n} A059365(n, k)*2^(n-k). - _Philippe Deléham_, Jan 19 2004
%F G.f.: 1/(1-x/(1-2x/(1-2x/(1-2x/(1-.... = 1/(1-x-2x^2/(1-4x-4x^2/(1-4x-4x^2/(1-.... (continued fractions). - _Paul Barry_, Jan 30 2009
%F a(n) = (32/Pi)*Integral_{x = 0..1} (8*x)^(n-1)*sqrt(x*(1-x)) / (8*x+1). - _Groux Roland_, Dec 12 2010
%F a(n+2) = 8^(n+2)*( c(n+2)-c(1)*c(n+1) - Sum_{i=0..n-1} 8^(-i-2)*c(n-i)*a(i+2) ) with c(n) = Catalan(n+2)/2^(2*n+1). - _Groux Roland_, Dec 12 2010
%F a(n) = the upper left term in M^n, M = the production matrix:
%F 1, 1
%F 2, 2, 1
%F 4, 4, 2, 1
%F 8, 8, 4, 2, 1
%F ... - _Gary W. Adamson_, Jul 08 2011
%F D-finite with recurrence: n*a(n) + (12-7n)*a(n-1) + 4*(3-2n)*a(n-2) = 0. - _R. J. Mathar_, Nov 16 2011 (This follows easily from the generating function. - _Robert Israel_, Nov 30 2014)
%F G.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^2 dx. - _Paul D. Hanna_, Dec 24 2013
%F G.f. satisfies: Integral 1/A(x)^2 dx = x - x^2*G(x), where G(x) is the o.g.f. of A000257, the number of rooted bicubic maps. - _Paul D. Hanna_, Dec 24 2013
%F G.f. A(x) satisfies: A(x - 2*x^2) = 1/(1-x). - _Paul D. Hanna_, Nov 30 2014
%F a(n) = hypergeometric([1-n, n], [-n], 2) for n > 0. - _Peter Luschny_, Nov 30 2014
%F G.f.: (3 - sqrt(1-8*x))/(2*(x+1)). - _Robert Israel_, Nov 30 2014
%F a(n) ~ 2^(3*n+1) / (9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Dec 22 2014
%F O.g.f. A(x) = 1 + series reversion of (x*(1 - x)/(1 + x)^2). Logarithmically differentiating (A(x) - 1)/x gives 3 + 17*x + 111*x^2 + ..., essentially a g.f for A119259. - _Peter Bala_, Oct 01 2015
%F From _Peter Bala_, Jan 06 2022: (Start)
%F exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + ... is a g.f. for A022558.
%F The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
%p 1, seq(simplify(hypergeom([1-n,n],[-n],2)), n=1..100); # _Robert Israel_, Nov 30 2014
%t a[0]=1; a[1]=1; a[n_]/;n>=2 := a[n] = a[n-1] + Sum[(a[k] + a[k-1])a[n-k],{k,n-1}]; Table[a[n],{n,0,10}] (* _David Callan_, Aug 27 2009 *)
%t a[n_] := 2*Sum[ (-1)^j*2^(n-j-1)*Binomial[2*(n-j-1), n-j-1]/(n-j), {j, 0, n-1}] + (-1)^n; Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jul 03 2013 *)
%o (PARI) {a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x),n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^2+x*O(x^n)))); polcoeff(A, n)} \\ _Paul D. Hanna_, Dec 24 2013
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=polcoeff(1/(1 - serreverse(x-2*x^2 +x^2*O(x^n))),n)}
%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 30 2014
%o (Sage)
%o def a(n):
%o if n==0: return 1
%o return hypergeometric([1-n, n], [-n], 2).simplify()
%o [a(n) for n in range(22)] # _Peter Luschny_, Dec 01 2014
%Y Cf. A064334, A064311, A119529.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Sep 13 2001
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