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A025035 Number of partitions of { 1, 2, ..., 3n } into sets of size 3. 46
1, 1, 10, 280, 15400, 1401400, 190590400, 36212176000, 9161680528000, 2977546171600000, 1208883745669600000, 599606337852121600000, 356765771022012352000000, 250806337028474683456000000, 205661196363349240433920000000, 194555491759728381450488320000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row sums of A157703. - Johannes W. Meijer, Mar 07 2009
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
REFERENCES
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.
LINKS
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Murray R. Bremner and Hader A. Elgendy, Special Identities for Comtrans Algebras, arXiv:1806.10204 [math.RA], 2018.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 17.
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. - Paul Barry, Sep 02 2010
J. Harmse and J. Remmel, Patterns in column strict fillings of rectangular arrays, Pure Mathematics and Applications, 22 (2011), 131-171. - Ran Pan, Apr 10 2015
Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, On certain combinatorial expansions of the Legendre-Stirling numbers, arXiv:1805.10998 [math.CO], 2018.
Ran Pan, Exercise J, Project P.
FORMULA
a(n) = (3*n)!/(n!*(3!)^n). - Christian G. Bower, Sep 01 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), 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
a(n) ~ sqrt(3)*9^n*n^(2*n)/(2^n*exp(2*n)). - Ilya Gutkovskiy, Aug 12 2016
a(n) = Pochhammer(n + 1, 2*n)/6^n. - Peter Luschny, Nov 18 2019
EXAMPLE
G.f. = 1 + x + 10*x^2 + 280*x^3 + 15400*x^4 + 1401400*x^5 + ...
MAPLE
a := pochhammer(n+1, 2*n)/6^n: seq(a(n), n=0..15); # Peter Luschny, Nov 18 2019
MATHEMATICA
Select[Range[0, 39]! CoefficientList[Series[Exp[x^3/3!], {x, 0, 39}], x], # > 0 &] (* Geoffrey Critzer, Sep 24 2011 *)
Table[(3 n)!/(n! (3!)^n), {n, 0, 15}] (* Michael De Vlieger, Aug 14 2016 *)
a[ n_] := With[{m = 3 n}, If[ m < 0, 0, m! SeriesCoefficient[Exp[x^3/3!], {x, 0, m}]]]; (* Michael Somos, Nov 25 2016 *)
PROG
(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
CROSSREFS
Column k=3 of A060540.
Sequence in context: A205824 A251580 A165457 * A012243 A186270 A231793
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 21 05:34 EDT 2024. Contains 372728 sequences. (Running on oeis4.)