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A025035
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Number of partitions of { 1, 2, ..., 3n } into sets of size 3.
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17
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1, 1, 10, 280, 15400, 1401400, 190590400, 36212176000, 9161680528000, 2977546171600000, 1208883745669600000, 599606337852121600000, 356765771022012352000000, 250806337028474683456000000, 205661196363349240433920000000, 194555491759728381450488320000000
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OFFSET
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0,3
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COMMENTS
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Row sums of A157703. - Johannes W. Meijer, Mar 07 2009
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REFERENCES
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P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Laughlin's wave functions, Coulomb gases and expansions of the discriminant, Int. J. Mod. Phys. A9 (1994) 4257. [From Paul Barry, Sep 02 2010]
B.G. Wybourne, Admissible partitions and the square of the Vandermonde determinant, 2003. [From Paul Barry, Sep 02 2010]
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LINKS
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Table of n, a(n) for n=0..15.
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FORMULA
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a(n) = (3*n)!/(n!*(3!)^n). (Christian G. Bower bowerc(AT)usa.net 9/1998).
Integral representation as n-th moment of a positive function on the positive axis, in Maple notation: int(x^n*sqrt(2/(3*x))*BesselK(1/3, 2*sqrt(2*x)/3)/Pi, x=0..infinity), n=0, 1... . Karol A. Penson, Oct 05 2005.
E.g.f.: exp(x^3/3!) (with interpolated zeros) - Paul Barry, May 26 2003
product( binomial(3*n-3*i,3) , i=0..n-1) / n! (equivalent to Christian Bower formula) - Olivier Gérard (olivier.gerard(AT)gmail.com, Feb 14 2011.
2*a(n) -(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Dec 03 2012
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MATHEMATICA
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Select[Range[0, 39]! CoefficientList[Series[Exp[x^3/3!], {x, 0, 39}], x], # > 0 &] (* Geoffrey Critzer, Sep 24 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, (3*n)! / n! / 6^n)} /* Michael Somos, Mar 26 2003 */
(PARI) {a(n) = if( n<0, 0, prod( i=0, n-1, binomial( 3*n - 3*i, 3)) / n!)} /* Michael Somos, Feb 15 2011 */
(Sage) [rising_factorial(n+1, 2*n)/6^n for n in (0..15)] # Peter Luschny, Jun 26 2012
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CROSSREFS
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Sequence in context: A157713 A205824 A165457 * A012243 A186270 A213403
Adjacent sequences: A025032 A025033 A025034 * A025036 A025037 A025038
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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STATUS
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approved
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