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
KEYWORD
nonn
AUTHOR
STATUS
approved