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A007054
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Super ballot numbers: 6(2n)!/(n!(n+2)!).
(Formerly M2243)
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15
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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
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OFFSET
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0,1
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COMMENTS
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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]
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REFERENCES
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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, 2012. - From N. J. A. Sloane, Oct 10 2012
I. M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Callan, A combinatorial interpretation for a super-Catalan recurrence
Ira Gessel, Rational functions with nonnegative power series, (slides).
Ira Gessel, Super ballot numbers.
Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of The Numbers 6(2n)! /n! (n+2)!, arXiv:math/0401300v2 [math.CO]
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. Schaeļ¬er, A combinatorial interpretation of super-Catalan numbers of order two, (2001).
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FORMULA
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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 DELEHAM, 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) [From 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
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MAPLE
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seq(3*(2*n)!/(n!)^2/binomial(n+2, n), n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007
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MATHEMATICA
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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] (* From Harvey P. Dale, Oct 05 2011 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A002421.
Cf. A007272.
Cf. A091712, A000257.
Sequence in context: A215413 A049923 A184881 * A084388 A136389 A141863
Adjacent sequences: A007051 A007052 A007053 * A007055 A007056 A007057
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Mira Bernstein, Ira M. Gessel
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EXTENSIONS
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Corrected and extended by Vincenzo Librandi, Aug 20 2011
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STATUS
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approved
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