A014606 a(n) = (3n)!/(6^n).

1, 1, 20, 1680, 369600, 168168000, 137225088000, 182509367040000, 369398958888960000, 1080491954750208000000, 4386797336285844480000000, 23934366266775567482880000000, 170891375144777551827763200000000, 1561776277448122046153927884800000000

Offset: 0

Author

BjornE (mdeans(AT)algonet.se)

Keywords

• nonn
• easy

Comments

a(n) is also the constant term in the product : product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^3. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 14 2002

a(n) is also the number of n by 3n (0,1)-matrices with row sum 3 and column sum 1. In general, the number of n by s*n (0,1)-matrices with row sum s and column sum 1 is (s*n)!/(s!)^n. - Shanzhen Gao, Feb 12 2010

References

• George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University Press, 1998.
• Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer., Vol. 202 (2010), pp. 45-53.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..165
• J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Crossrefs

Cf. A000680, A025035, A089759.

Programs

Mathematica

nn=36;Select[Range[0,nn]!CoefficientList[Series[1/(1-x^3/3!),{x,0,nn}],x],#>0&] (* Geoffrey Critzer, Jun 07 2014 *)

PARI

a(n)=(3*n)!/6^n;

Formula

E.g.f. with interpolated zeros: 1/(1 - x^3/3!). - Geoffrey Critzer, Jun 07 2014

a(n) = A025035(n)*n! - Geoffrey Critzer, Jun 07 2014

a(n) = A089759(3,n). - R. J. Mathar, Nov 01 2015

From Amiram Eldar, Jan 26 2022: (Start)

Sum_{n>=0} 1/a(n) = (exp(6^(1/3)) + 2*exp(-6^(1/3)/2)*cos(3^(5/6)/2^(2/3)))/3.

Sum_{n>=0} (-1)^n/a(n) = (exp(-6^(1/3)) + 2*exp(6^(1/3)/2)*cos(3^(5/6)/2^(2/3)))/3. (End)