0,1

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

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

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.

D. Callan, A combinatorial interpretation for a super-Catalan recurrence

Ira Gessel, Rational functions with nonnegative power series, (slides).

Ira Gessel, Super ballot numbers.

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

seq(3*(2*n)!/(n!)^2/binomial(n+2, n), n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007

Cf. A002421.

Cf. A007272.

Cf. A091712, A000257.

Sequence in context: A058533 A058644 A049923 this_sequence A084388 A136389 A001368

Adjacent sequences: A007051 A007052 A007053 this_sequence A007055 A007056 A007057

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Ira Gessel