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A007054 Super ballot numbers: 6(2n)!/(n!(n+2)!).
(Formerly M2243)
16
3, 2, 3, 6, 14, 36, 99, 286, 858, 2652, 8398, 27132, 89148, 297160, 1002915, 3421710, 11785890, 40940460, 143291610, 504932340, 1790214660, 6382504440, 22870640910, 82334307276, 297670187844, 1080432533656, 3935861372604, 14386251913656, 52749590350072 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Hankel transform is 2n+3. The Hankel transform of a(n+1) is n+2. The sequence a(n)-2*0^n has Hankel transform A110331(n). - Paul Barry, Jul 20 2008

Number of pairs of Dyck paths of total length 2*n with heights differing by at most 1. (Gessel/Xin, p.2). - Joerg Arndt, Sep 01 2012

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

E. Allen, I. Gheorghiciuc, A Weighted Interpretation for the Super Catalan Numbers, J. Int. Seq. 17 (2014) # 14.10.7.

D. Callan, A combinatorial interpretation for a super-Catalan recurrence, arXiv:math/0408117 [math.CO], 2004.

David Callan, A Combinatorial Interpretation for a Super-Catalan Recurrence, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.8.

D. Callan, A variant of Touchard's Catalan number identity, arXiv preprint arXiv:1204.5704 [math.CO], 2012. - From N. J. A. Sloane, Oct 10 2012

Ira M. Gessel, Letter to N. J. A. Sloane, Jul. 1992

Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194

Ira M. Gessel, Rational Functions With Nonnegative Integer Coefficients, 50th Séminaire Lotharingien de Combinatoire, 2003.

Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of the Numbers 6(2n)!/n!(n+2)!, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3.

Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of The Numbers 6(2n)! /n! (n+2)!, arXiv:math/0401300v2 [math.CO], 2004.

N. Pippenger and K. Schleich, Topological characteristics of random triangulated surfaces (section 7), Random Structures Algorithms 28 (2006) 247-288; arXiv:gr-qc/0306049v1.

G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).

FORMULA

G.f.: c(x)*(4-c(x)), where c(x) = g.f. for Catalan numbers A000108; Convolution of Catalan numbers with negative Catalan numbers but -C(0)=-1 replaced by 3. - Wolfdieter Lang

E.g.f. in Maple notation: exp(2*x)*(4*x*(BesselI(0, 2*x)-BesselI(1, 2*x))-BesselI(1, 2*x))/x. Integral representation as n-th moment of a positive function on [0, 4], in Maple notation: a(n)=int(x^n*(4-x)^(3/2)/x^(1/2), x=0..4)/(2*Pi), n=0, 1... This representation is unique. - Karol A. Penson, Oct 10 2001

E.g.f.: sum(n>=0, a(n)*x^(2*n) ) = 3*BesselI(2, 2x).

a(n) = A000108(n)*6/(n+2). - Philippe Deléham, Oct 30 2007

a(n+1) = 2*(A000108(n+2)-A000108(n+1))/(n+1); - Paul Barry, Jul 20 2008

G.f.: ((6-4*sqrt(1-4*x))*x+sqrt(1-4*x)-1)/(2*x^2) - Harvey P. Dale, Oct 05 2011

a(n) = 4*A000108(n) - A000108(n+1) (Gessel/Xin, p. 2). - Joerg Arndt, Sep 01 2012

(n+2)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Dec 03 2012

G.f.: 1/(x^2*G(0)) + 3/x -1/2/x^2, where G(k)= 1 + 1/(1 - 2*x*(2*k+3)/(2*x*(2*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

G.f.: 3/x - 1/(2*x^2) + G(0)/(4*x^2), where G(k)= 1 + 1/(1 - 2*x*(2*k-3)/(2*x*(2*k-3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013

0 = a(n)*(+16*a(n+1) - 14*a(n+2)) + a(n+1)*(+6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 18 2014

A002421(n+2) = 2*a(n) for all n in Z. - Michael Somos, Sep 18 2014

a(n) = 3*(2*n)!*[x^(2*n)]hypergeometric([],[3],x^2). - Peter Luschny, Feb 01 2015

a(n) = 6*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(3+n)). - Peter Luschny, Dec 14 2015

MAPLE

seq(3*(2*n)!/(n!)^2/binomial(n+2, n), n=0..22); # Zerinvary Lajos, Jun 28 2007

A007054 := n -> 6*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(3+n)):

seq(A007054(n), n=0..28); # Peter Luschny, Dec 14 2015

MATHEMATICA

Table[6(2n)!/(n!(n+2)!), {n, 0, 30}] (* or *) CoefficientList[Series[ (-1+Sqrt[1-4*x]+(6-4*Sqrt[1-4*x])*x)/(2*x^2), {x, 0, 30}], x] (* Harvey P. Dale, Oct 05 2011 *)

PROG

(MAGMA) [6*Factorial(2*n)/(Factorial(n)*Factorial(n+2)): n in [0..30]]; // Vincenzo Librandi, Aug 20 2011

(PARI) a(n)=6*(2*n)!/(n!*(n+2)!); /* Joerg Arndt, Sep 01 2012 */

CROSSREFS

Cf. A002421, A007272, A091712, A000257.

Sequence in context: A058644 A049923 A184881 * A084388 A136389 A275330

Adjacent sequences:  A007051 A007052 A007053 * A007055 A007056 A007057

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mira Bernstein, Ira M. Gessel

EXTENSIONS

Corrected and extended by Vincenzo Librandi, Aug 20 2011

STATUS

approved

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Last modified May 26 04:46 EDT 2019. Contains 323579 sequences. (Running on oeis4.)