%I A007054 M2243
%S A007054 3,2,3,6,14,36,99,286,858,2652,8398,27132,89148,297160,1002915,3421710,
%T A007054 11785890,40940460,143291610,504932340,1790214660,6382504440,22870640910
%N A007054 Super ballot numbers: 6(2n)!/n!(n+2)!.
%C A007054 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
%D A007054 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007054 David Callan, A Combinatorial Interpretation for a Super-Catalan Recurrence, 
               Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.8.
%D A007054 I. M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.
%D A007054 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.
%H A007054 D. Callan, <a href="http://arXiv.org/abs/math.CO/0408117">A combinatorial 
               interpretation for a super-Catalan recurrence</a>
%H A007054 Ira Gessel, <a href="http://www.cs.brandeis.edu/~ira/">Rational functions 
               with nonnegative power series</a>, (slides).
%H A007054 Ira Gessel, <a href="http://www.cs.brandeis.edu/~ira/">Super ballot numbers</
               a>.
%F A007054 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.
%F A007054 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
%F A007054 E.g.f.: Sum[n>=0, a(n)*x^(2n)] = 3*BesselI(2, 2x).
%F A007054 a(n)=A000108(n)*6/(n+2). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 30 2007
%F A007054 a(n+1)=2*(A000108(n+2)-A000108(n+1))/(n+1); - Paul Barry (pbarry(AT)wit.ie), 
               Jul 20 2008
%p A007054 seq(3*(2*n)!/(n!)^2/binomial(n+2,n), n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 28 2007
%Y A007054 Cf. A002421.
%Y A007054 Cf. A007272.
%Y A007054 Cf. A091712, A000257.
%Y A007054 Sequence in context: A058533 A058644 A049923 this_sequence A084388 A136389 
               A001368
%Y A007054 Adjacent sequences: A007051 A007052 A007053 this_sequence A007055 A007056 
               A007057
%K A007054 nonn,easy
%O A007054 0,1
%A A007054 N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Ira Gessel