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

0,3

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

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

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.

Cf. A001147, A025036.

Sequence in context: A205824 A251580 A165457 * A012243 A186270 A231793

Adjacent sequences: A025032 A025033 A025034 * A025036 A025037 A025038

Keyword

nonn

Author

David W. Wilson

Status

approved