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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

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

MAPLE

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

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

N. J. A. Sloane, Simon Plouffe, Ira M. Gessel

STATUS

approved

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Last modified December 3 10:38 EST 2016. Contains 278699 sequences.