A064062 Generalized Catalan numbers C(2; n).

1, 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

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.

Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.

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.

S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

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[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 *)
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: A365250 A200754 A062992 * A114191 A107592 A215257

Adjacent sequences: A064059 A064060 A064061 * A064063 A064064 A064065

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 13 2001

Status

approved