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 A007272 Super ballot numbers: 60(2n)!/(n!(n+3)!). (Formerly M4676) 11
 10, 5, 6, 10, 20, 45, 110, 286, 780, 2210, 6460, 19380, 59432, 185725, 589950, 1900950, 6203100, 20470230, 68234100, 229514700, 778354200, 2659376850, 9148256364, 31667041260, 110248217720, 385868762020, 1357193576760, 4795417304552, 17015996887120, 60619488910365 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Matthew House, Table of n, a(n) for n = 0..1677 E. Allen, I. Gheorghiciuc, A Weighted Interpretation for the Super Catalan Numbers, J. Int. Seq. 17 (2014) # 14.10.7. D. Callan, A combinatorial interpretation for a super-Catalan recurrence, arXiv:math/0408117 [math.CO], 2004. 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. FORMULA 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). This representation is unique. - Karol A. Penson, Dec 04 2001 a(n) = 10*(2*n)!*[x^(2*n)](hypergeometric([],[4],x^2)). - Peter Luschny, Feb 01 2015 (n+3)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 06 2018 MAPLE seq(10*(2*n)!/(n!)^2/binomial(n+3, n), n=0..26); # Zerinvary Lajos, Jun 28 2007 MATHEMATICA Table[60(2n)!/(n!(n+3)!), {n, 0, 30}] (* Jean-François Alcover, Jun 02 2019 *) PROG (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 */ CROSSREFS Cf. A002422. Sequence in context: A066578 A097327 A226583 * A061280 A030071 A147653 Adjacent sequences:  A007269 A007270 A007271 * A007273 A007274 A007275 KEYWORD nonn AUTHOR STATUS approved

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Last modified June 17 17:07 EDT 2019. Contains 324196 sequences. (Running on oeis4.)