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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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46
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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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Number of bottom-row-increasing column-strict arrays of size 3 X n. - Ran Pan, Apr 10 2015
a(n) is the number of rooted semi-labeled or phylogenetic trees with n interior vertices and each interior vertex having out-degree 3. - Albert Alejandro Artiles Calix, Aug 12 2016
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REFERENCES
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Erdos, Peter L., and L A. Szekely. "Applications of antilexicographic order. I. An enumerative theory of trees." Academic Press Inc. (1989): 488-96. Web. 4 July 2016.
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LINKS
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FORMULA
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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), for n>=0. - Karol A. Penson, Oct 05 2005
E.g.f.: exp(x^3/3!) (with interpolated zeros). - Paul Barry, May 26 2003
a(n) = Product_{i=0..n-1} binomial(3*n-3*i,3) / n! (equivalent to Christian Bower formula). - Olivier Gérard, 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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EXAMPLE
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G.f. = 1 + x + 10*x^2 + 280*x^3 + 15400*x^4 + 1401400*x^5 + ...
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MAPLE
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a := pochhammer(n+1, 2*n)/6^n: seq(a(n), n=0..15); # Peter Luschny, Nov 18 2019
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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 *)
a[ n_] := With[{m = 3 n}, If[ m < 0, 0, m! SeriesCoefficient[Exp[x^3/3!], {x, 0, m}]]]; (* Michael Somos, Nov 25 2016 *)
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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
(Magma) [Factorial(3*n)/(Factorial(n)*6^n): n in [0..20]]; // Vincenzo Librandi, Apr 10 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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