A025035 Number of partitions of { 1, 2, ..., 3n } into sets of size 3.

1, 1, 10, 280, 15400, 1401400, 190590400, 36212176000, 9161680528000, 2977546171600000, 1208883745669600000, 599606337852121600000, 356765771022012352000000, 250806337028474683456000000, 205661196363349240433920000000, 194555491759728381450488320000000

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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

Examples

			G.f. = 1 + x + 10*x^2 + 280*x^3 + 15400*x^4 + 1401400*x^5 + ...
		

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

• Alois P. Heinz, Table of n, a(n) for n = 0..220
• 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.
• Peter L. Erdos and L. A. Szekelly, Applications of antilexicographic order. I. An enumerative theory of trees.
• 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.
• B. G. Wybourne, Admissible partitions and the square of the Vandermonde determinant, 2003. - _Paul Barry_, Sep 02 2010

Crossrefs

Column k=3 of A060540.

Cf. A001147, A025036.

Programs

Magma

[Factorial(3*n)/(Factorial(n)*6^n): n in [0..20]]; // Vincenzo Librandi, Apr 10 2015

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 *)

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

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: a(n) = Integral_{x>=0} x^n*sqrt(2/(3*x))*BesselK(1/3, 2*sqrt(2*x)/3)/Pi dx, 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