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A007054
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Super ballot numbers: 6(2n)!/(n!(n+2)!).
(Formerly M2243)
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14
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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 (pbarry(AT)wit.ie), Jul 20 2008
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REFERENCES
| David Callan, A Combinatorial Interpretation for a Super-Catalan Recurrence, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.8.
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
| 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.
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
| Comment from wolfdieter.lang(AT)physik.uni-karlsruhe.de: 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.
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 (penson(AT)lptl.jussieu.fr), Oct 10 2001
E.g.f.: Sum[n>=0, a(n)*x^(2n)] = 3*BesselI(2, 2x).
a(n)=A000108(n)*6/(n+2). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2007
a(n+1)=2*(A000108(n+2)-A000108(n+1))/(n+1); - Paul Barry (pbarry(AT)wit.ie), 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]
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MAPLE
| 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
| 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
| (MAGMA) [6*Factorial(2*n)/(Factorial(n)*Factorial(n+2)): n in [0..30]]; // Vincenzo Librandi, Aug 20 2011
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CROSSREFS
| Cf. A002421.
Cf. A007272.
Cf. A091712, A000257.
Sequence in context: A058644 A049923 A184881 * A084388 A136389 A001368
Adjacent sequences: A007051 A007052 A007053 * A007055 A007056 A007057
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Ira Gessel
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EXTENSIONS
| Corrected and extended by Vincenzo Librandi, Aug 20 2011
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