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 A064062 Generalized Catalan numbers C(2; n). 28
 1, 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). 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 The Hankel transform of this sequence is A002416. - Philippe Deléham, Nov 19 2007 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 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3. Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From N. J. A. Sloane, Oct 08 2012 J. Bloom and S. Elizalde, Pattern avoidance in matchings and partitions, arXiv:1211.3442 [math.CO] (2012) Theorem 6.1. N. Bonichon, C. Gavoille and N. Hanusse, Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation, In Proceedings of WG'03, volume 2880 of LNCS, pp. 81-92, 2003. Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006. Xiang-Ke Chang, X.-B. Hu, H. Lei and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8. Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3. N. J. A. Sloane, Transforms A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011-2012. FORMULA 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. 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. a(n) = Sum_{k = 0..n} A059365(n, k)*2^(n-k). - Philippe Deléham, Jan 19 2004 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 a(n) = (32/Pi)*Integral_{x = 0..1} (8*x)^(n-1)*sqrt(x*(1-x)) / (8*x+1). - Groux Roland, Dec 12 2010 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 a(n) = the upper left term in M^n, M = the production matrix:   1, 1   2, 2, 1   4, 4, 2, 1   8, 8, 4, 2, 1   ... - Gary W. Adamson, Jul 08 2011 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) G.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^2 dx. - Paul D. Hanna, Dec 24 2013 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 G.f. A(x) satisfies: A(x - 2*x^2) = 1/(1-x). - Paul D. Hanna, Nov 30 2014 a(n) = hypergeometric([1-n, n], [-n], 2) for n > 0. - Peter Luschny, Nov 30 2014 G.f.: (3 - sqrt(1-8*x))/(2*(x+1)). - Robert Israel, Nov 30 2014 a(n) ~ 2^(3*n+1) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 22 2014 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 From Peter Bala, Jan 06 2022: (Start) 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. 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) MAPLE 1, seq(simplify(hypergeom([1-n, n], [-n], 2)), n=1..100); # Robert Israel, Nov 30 2014 MATHEMATICA a=1; a=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 *) 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 *) PROG (PARI) {a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x), n)} (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 for(n=0, 25, print1(a(n), ", ")) (PARI) {a(n)=polcoeff(1/(1 - serreverse(x-2*x^2 +x^2*O(x^n))), n)} for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 30 2014 (Sage) def a(n):     if n==0: return 1     return hypergeometric([1-n, n], [-n], 2).simplify() [a(n) for n in range(22)] # Peter Luschny, Dec 01 2014 CROSSREFS Cf. A064334, A064311, A119529. Sequence in context: A234282 A200754 A062992 * A114191 A107592 A215257 Adjacent sequences:  A064059 A064060 A064061 * A064063 A064064 A064065 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Sep 13 2001 STATUS approved

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Last modified August 8 10:26 EDT 2022. Contains 356009 sequences. (Running on oeis4.)