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A007272
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Super ballot numbers: 60(2n)!/(n!(n+3)!).
(Formerly M4676)
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10
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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
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OFFSET
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0,1
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REFERENCES
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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
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Table of n, a(n) for n=0..26.
D. Callan, A combinatorial interpretation for a super-Catalan recurrence
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FORMULA
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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, Dec 04 2001
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MAPLE
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seq(10*(2*n)!/(n!)^2/binomial(n+3, n), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007
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PROG
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(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 */
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CROSSREFS
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Cf. A002422.
Sequence in context: A080461 A066578 A097327 * A061280 A030071 A147653
Adjacent sequences: A007269 A007270 A007271 * A007273 A007274 A007275
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Simon Plouffe, Ira M. Gessel
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STATUS
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approved
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