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%I #106 May 26 2023 01:03:09
%S 1,1,10,280,15400,1401400,190590400,36212176000,9161680528000,
%T 2977546171600000,1208883745669600000,599606337852121600000,
%U 356765771022012352000000,250806337028474683456000000,205661196363349240433920000000,194555491759728381450488320000000
%N Number of partitions of { 1, 2, ..., 3n } into sets of size 3.
%C Row sums of A157703. - _Johannes W. Meijer_, Mar 07 2009
%C Number of bottom-row-increasing column-strict arrays of size 3 X n. - _Ran Pan_, Apr 10 2015
%C 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
%D 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.
%H Alois P. Heinz, <a href="/A025035/b025035.txt">Table of n, a(n) for n = 0..220</a>
%H Cyril Banderier, Philippe Marchal, and Michael Wallner, <a href="https://arxiv.org/abs/1805.09017">Rectangular Young tableaux with local decreases and the density method for uniform random generation</a> (short version), arXiv:1805.09017 [cs.DM], 2018.
%H Murray R. Bremner and Hader A. Elgendy, <a href="https://arxiv.org/abs/1806.10204">Special Identities for Comtrans Algebras</a>, arXiv:1806.10204 [math.RA], 2018.
%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 17.
%H Peter L. Erdos and L. A. Szekelly, <a href="https://www.researchgate.net/publication/256513203_Applications_of_antilexicographic_order_I_An_enumerative_theory_of_trees">Applications of antilexicographic order. I. An enumerative theory of trees.</a>
%H P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, <a href="http://dx.doi.org/10.1142/S0217751X94001734">Laughlin's wave functions, Coulomb gases and expansions of the discriminant</a>, Int. J. Mod. Phys. A9 (1994) 4257. - _Paul Barry_, Sep 02 2010
%H J. Harmse and J. Remmel, <a href="http://www.mat.unisi.it/newsito/puma/public_html/22_2/harmse_remmel.pdf">Patterns in column strict fillings of rectangular arrays</a>, Pure Mathematics and Applications, 22 (2011), 131-171. - _Ran Pan_, Apr 10 2015
%H Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, <a href="https://arxiv.org/abs/1805.10998">On certain combinatorial expansions of the Legendre-Stirling numbers</a>, arXiv:1805.10998 [math.CO], 2018.
%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eJ.html">Exercise J</a>, Project P.
%H B. G. Wybourne, <a href="http://www.fizyka.umk.pl/~bgw/bgwybourne.pdf">Admissible partitions and the square of the Vandermonde determinant</a>, 2003. - _Paul Barry_, Sep 02 2010
%F a(n) = (3*n)!/(n!*(3!)^n). - _Christian G. Bower_, Sep 01 1998
%F 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
%F E.g.f.: exp(x^3/3!) (with interpolated zeros). - _Paul Barry_, May 26 2003
%F a(n) = Product_{i=0..n-1} binomial(3*n-3*i,3) / n! (equivalent to Christian Bower formula). - _Olivier Gérard_, Feb 14 2011
%F 2*a(n) - (3*n-1)*(3*n-2)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012
%F a(n) ~ sqrt(3)*9^n*n^(2*n)/(2^n*exp(2*n)). - _Ilya Gutkovskiy_, Aug 12 2016
%F a(n) = Pochhammer(n + 1, 2*n)/6^n. - _Peter Luschny_, Nov 18 2019
%e G.f. = 1 + x + 10*x^2 + 280*x^3 + 15400*x^4 + 1401400*x^5 + ...
%p a := pochhammer(n+1, 2*n)/6^n: seq(a(n), n=0..15); # _Peter Luschny_, Nov 18 2019
%t Select[Range[0, 39]! CoefficientList[Series[Exp[x^3/3!], {x, 0, 39}], x], # > 0 &] (* _Geoffrey Critzer_, Sep 24 2011 *)
%t Table[(3 n)!/(n! (3!)^n), {n, 0, 15}] (* _Michael De Vlieger_, Aug 14 2016 *)
%t a[ n_] := With[{m = 3 n}, If[ m < 0, 0, m! SeriesCoefficient[Exp[x^3/3!], {x, 0, m}]]]; (* _Michael Somos_, Nov 25 2016 *)
%o (PARI) {a(n) = if( n<0, 0, (3*n)! / n! / 6^n)}; /* _Michael Somos_, Mar 26 2003 */
%o (PARI) {a(n) = if( n<0, 0, prod( i=0, n-1, binomial( 3*n - 3*i, 3)) / n!)}; /* _Michael Somos_, Feb 15 2011 */
%o (Sage) [rising_factorial(n+1,2*n)/6^n for n in (0..15)] # _Peter Luschny_, Jun 26 2012
%o (Magma) [Factorial(3*n)/(Factorial(n)*6^n): n in [0..20]]; // _Vincenzo Librandi_, Apr 10 2015
%Y Column k=3 of A060540.
%Y Cf. A001147, A025036.
%K nonn
%O 0,3
%A _David W. Wilson_
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