0,1

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

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

G.f.: (11-32*x+9*sqrt(1-4*x))/(1-3*x+(1-x)*sqrt(1-4*x)).

E.g.f.: Sum_{n>=0} a(n)*x^(2n)/(2n)! = 60*BesselI(3, 2x)/x^3.

E.g.f.: (BesselI(0, 2*x)*(2*x+16*x^2)-BesselI(1, 2*x)*(2+6*x+16*x^2))*exp(2*x)/x^2.

Integral representation as the n-th moment of a positive function on [0, 4], in Maple notation : a(n)=int(x^n*1/2*(4-x)^(5/2)/Pi/x^(1/2), x=0..4), n=0, 1, ... . This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Dec 04 2001

seq(10*(2*n)!/(n!)^2/binomial(n+3, n), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007

(PARI) a(n)=if(n<0, 0, 60*(2*n)!/n!/(n+3)!) /* Michael Somos Feb 19 2006 */

(PARI) {a(n)=if(n<0, 0, n*=2; n!*polcoeff( 10*besseli(3, 2*x+x*O(x^n)), n))} /* Michael Somos Feb 19 2006 */

Cf. A002422.

Sequence in context: A080461 A066578 A097327 this_sequence A061280 A030071 A147653

Adjacent sequences: A007269 A007270 A007271 this_sequence A007273 A007274 A007275

nonn

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Ira Gessel