%I M2243 #113 Jul 12 2024 05:17:36
%S 3,2,3,6,14,36,99,286,858,2652,8398,27132,89148,297160,1002915,
%T 3421710,11785890,40940460,143291610,504932340,1790214660,6382504440,
%U 22870640910,82334307276,297670187844,1080432533656,3935861372604,14386251913656,52749590350072
%N Super ballot numbers: 6(2n)!/(n!(n+2)!).
%C 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
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007054/b007054.txt">Table of n, a(n) for n = 0..200</a>
%H E. Allen and I. Gheorghiciuc, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Allen/gheo.html">A Weighted Interpretation for the Super Catalan Numbers</a>, J. Int. Seq. 17 (2014) # 14.10.7.
%H David Callan, <a href="https://arxiv.org/abs/math/0408117">A combinatorial interpretation for a super-Catalan recurrence</a>, arXiv:math/0408117 [math.CO], 2004.
%H David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Callan/callan301.html">A Combinatorial Interpretation for a Super-Catalan Recurrence</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.8.
%H David Callan, <a href="http://arxiv.org/abs/1204.5704">A variant of Touchard's Catalan number identity</a>, arXiv preprint arXiv:1204.5704 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 10 2012
%H Ira M. Gessel, <a href="/A007054/a007054.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>
%H Ira M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>, J. Symbolic Comp., 14 (1992), 179-194
%H Ira M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/slides/nonneg.pdf">Rational Functions With Nonnegative Integer Coefficients</a>, 50th Séminaire Lotharingien de Combinatoire, 2003.
%H Ira M. Gessel and Guoce Xin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Gessel/xin.html">A Combinatorial Interpretation of the Numbers 6(2n)!/n!(n+2)!</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3.
%H Ira M. Gessel and Guoce Xin, <a href="http://arxiv.org/abs/math/0401300">A Combinatorial Interpretation of The Numbers 6(2n)! /n! (n+2)!</a>, arXiv:math/0401300v2 [math.CO], 2004.
%H N. Pippenger and K. Schleich, <a href="http://arxiv.org/abs/gr-qc/0306049">Topological characteristics of random triangulated surfaces (section 7)</a>, Random Structures Algorithms 28 (2006) 247-288; arXiv:gr-qc/0306049v1.
%H G. Schaeffer, <a href="http://www.loria.fr/~schaeffe/Pub/Conjugacy/superCat.ps">A combinatorial interpretation of super-Catalan numbers of order two, (2001).
%F 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_
%F 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
%F E.g.f.: Sum_{n>=0} a(n)*x^(2*n) = 3*BesselI(2, 2x).
%F a(n) = A000108(n)*6/(n+2). - _Philippe Deléham_, Oct 30 2007
%F a(n+1) = 2*(A000108(n+2) - A000108(n+1))/(n+1). - _Paul Barry_, Jul 20 2008
%F G.f.: ((6-4*sqrt(1-4*x))*x+sqrt(1-4*x)-1)/(2*x^2) - _Harvey P. Dale_, Oct 05 2011
%F a(n) = 4*A000108(n) - A000108(n+1) (Gessel/Xin, p. 2). - _Joerg Arndt_, Sep 01 2012
%F D-finite with recurrence (n+2)*a(n) +2*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Dec 03 2012
%F 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
%F 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
%F 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
%F A002421(n+2) = 2*a(n) for all n in Z. - _Michael Somos_, Sep 18 2014
%F a(n) = 3*(2*n)!*[x^(2*n)]hypergeometric([],[3],x^2). - _Peter Luschny_, Feb 01 2015
%F a(n) = 6*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(3+n)). - _Peter Luschny_, Dec 14 2015
%F a(n) = (-4)^(2 + n)*binomial(3/2, 2 + n)/2. - _Peter Luschny_, Nov 04 2021
%F From _Amiram Eldar_, May 16 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1 + 20*Pi/(81*sqrt(3)).
%F Sum_{n>=0} (-1)^n/a(n) = 3/25 - 8*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). (End)
%F a(n-1) = 3*A000984(n)/((2*n-1)*(n+1)). - _R. J. Mathar_, Jul 12 2024
%p seq(3*(2*n)!/(n!)^2/binomial(n+2,n), n=0..22); # _Zerinvary Lajos_, Jun 28 2007
%p A007054 := n -> 6*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(3+n)):
%p seq(A007054(n),n=0..28); # _Peter Luschny_, Dec 14 2015
%t 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 *)
%o (Magma) [6*Factorial(2*n)/(Factorial(n)*Factorial(n+2)): n in [0..30]]; // _Vincenzo Librandi_, Aug 20 2011
%o (PARI) a(n)=6*(2*n)!/(n!*(n+2)!); /* _Joerg Arndt_, Sep 01 2012 */
%o (Sage)
%o def A007054(n): return (-4)^(2 + n)*binomial(3/2, 2 + n)/2
%o print([A007054(n) for n in range(29)]) # _Peter Luschny_, Nov 04 2021
%Y Cf. A002421, A007272, A091712, A000257, A001622.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, _Mira Bernstein_, _Ira M. Gessel_
%E Corrected and extended by _Vincenzo Librandi_, Aug 20 2011