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A007054
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Super ballot numbers: 6(2n)!/(n!(n+2)!).
(Formerly M2243)
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23
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3, 2, 3, 6, 14, 36, 99, 286, 858, 2652, 8398, 27132, 89148, 297160, 1002915, 3421710, 11785890, 40940460, 143291610, 504932340, 1790214660, 6382504440, 22870640910, 82334307276, 297670187844, 1080432533656, 3935861372604, 14386251913656, 52749590350072
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OFFSET
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0,1
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COMMENTS
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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, Jul 20 2008
Number of pairs of Dyck paths of total length 2*n with heights differing by at most 1 (Gessel/Xin, p. 2). - Joerg Arndt, Sep 01 2012
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REFERENCES
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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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FORMULA
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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. - Wolfdieter Lang
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, Oct 10 2001
E.g.f.: Sum_{n>=0} a(n)*x^(2*n) = 3*BesselI(2, 2x).
G.f.: ((6-4*sqrt(1-4*x))*x+sqrt(1-4*x)-1)/(2*x^2) - Harvey P. Dale, Oct 05 2011
D-finite with recurrence (n+2)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Dec 03 2012
G.f.: 1/(x^2*G(0)) + 3/x - (1/2)/x^2, where G(k) = 1 + 1/(1 - 2*x*(2*k+3)/(2*x*(2*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: 3/x - 1/(2*x^2) + G(0)/(4*x^2), where G(k) = 1 + 1/(1 - 2*x*(2*k-3)/(2*x*(2*k-3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013
0 = a(n)*(+16*a(n+1) - 14*a(n+2)) + a(n+1)*(+6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 18 2014
a(n) = 3*(2*n)!*[x^(2*n)]hypergeometric([],[3],x^2). - Peter Luschny, Feb 01 2015
a(n) = 6*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(3+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^(2 + n)*binomial(3/2, 2 + n)/2. - Peter Luschny, Nov 04 2021
Sum_{n>=0} 1/a(n) = 1 + 20*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 3/25 - 8*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). (End)
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MAPLE
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seq(3*(2*n)!/(n!)^2/binomial(n+2, n), n=0..22); # Zerinvary Lajos, Jun 28 2007
A007054 := n -> 6*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(3+n)):
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MATHEMATICA
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Table[6(2n)!/(n!(n+2)!), {n, 0, 30}] (* or *) CoefficientList[Series[ (-1+Sqrt[1-4*x]+(6-4*Sqrt[1-4*x])*x)/(2*x^2), {x, 0, 30}], x] (* Harvey P. Dale, Oct 05 2011 *)
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PROG
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(Magma) [6*Factorial(2*n)/(Factorial(n)*Factorial(n+2)): n in [0..30]]; // Vincenzo Librandi, Aug 20 2011
(PARI) a(n)=6*(2*n)!/(n!*(n+2)!); /* Joerg Arndt, Sep 01 2012 */
(Sage)
def A007054(n): return (-4)^(2 + n)*binomial(3/2, 2 + n)/2
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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