A006542 a(n) = binomial(n,3)*binomial(n-1,3)/4.

1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626, 3708810, 4562425, 5573800, 6765440, 8162176, 9791320, 11682825, 13869450

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of permutations of n+4 that avoid the pattern 132 and have exactly 3 descents. - Mike Zabrocki, Aug 26 2004

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 20 2005

a(n) = number of Dyck n-paths with exactly 4 peaks. - David Callan, Jul 03 2006

Six-dimensional figurate numbers for a hyperpyramid with pentagonal base. This corresponds to the sum(sum(sum(sum(1+sum(5*n))))) interpretation, see the Munafo webpage. - Robert Munafo, Jun 18 2009

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 166, no. 1).
• S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 4..200
• Isaac Ahern and Sam Cook, Affine Symmetry Tensors in Minkowski Space, American Journal of Undergraduate Research, Volume 13, Issue 3, August 2016.
• P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
• Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
• V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977
• G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
• G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 6, 25.
• Robert Munafo, C(n,3)C(n-1,3)/4
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces the sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.

Cf. A000332, A000579, A001263, A002378, A004068, A005585, A005891, A006322, A006414, A047819, A107891, A114242.

Fourth column of the table of Narayana numbers A001263.

Apart from a scale factor, a column of A124428.

Programs

GAP

List([4..40], n-> n*(n-1)^2*(n-2)^2*(n-3)/144); # G. C. Greubel, Feb 24 2019

Magma

[ n*((n-1)*(n-2))^2*(n-3)/144 : n in [4..40] ]; // Wesley Ivan Hurt, Jun 17 2014

Maple

A006542:=-(1+3*z+z**2)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation
A006542:=n->n*((n-1)*(n-2))^2*(n-3)/144; seq(A006542(n), n=4..40); # Wesley Ivan Hurt, Jun 17 2014

Mathematica

Table[Binomial[n, 3]*Binomial[n-1, 3]/4, {n, 4, 40}]

PARI

a(n)=n*((n-1)*(n-2))^2*(n-3)/144

Sage

[n*(n-1)^2*(n-2)^2*(n-3)/144 for n in (4..40)] # G. C. Greubel, Feb 24 2019

Formula

a(n) = C(n, 3)*C(n-1, 3)/4 = n*(n-1)^2*(n-2)^2*(n-3)/144.

a(n) = A000292(n-3)*A000292(n-2)/4.

E.g.f.: x^4*(6 + 6*x + x^2)*exp(x)/144. - Vladeta Jovovic, Jan 29 2003

a(n) = Sum(Sum(Sum(Sum(1 + Sum(5*n))))) = Sum (A006414). - Xavier Acloque, Oct 08 2003

a(n) = C(n, 6) + 3*C(n+1, 6) + C(n+2, 6). - Mike Zabrocki, Aug 26 2004

G.f.: x^4*(1 + 3*x + x^2)/(1-x)^7. - Emeric Deutsch, Jun 20 2005

a(n) = C(n-2, n-4)*C(n-1, n-3)*C(n, n-2)/18. - Zerinvary Lajos, Jul 29 2005

a(n) = C(n,4)*C(n,3)/n. - Mitch Harris, Jul 06 2006

a(n+2) = (1/4)*Sum_{1 <= x_1, x_2 <= n} x_1*x_2*(det V(x_1,x_2))^2 = (1/4)*Sum_{1 <= i,j <= n} i*j*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007

a(n) = C(n-1,3)^2 - C(n-1,2)*C(n-1,4). - Gary Detlefs, Dec 05 2011

a(n) = A000292(A000217(n-1)) - A000217(A000292(n-1)). - Ivan N. Ianakiev, Jun 17 2014

a(n) = Product_{i=1..3} A002378(n-4+i)/A002378(i). - Bruno Berselli, Nov 12 2014 (Rewritten, Sep 01 2016.)

Sum_{n>=4} 1/a(n) = 238 - 24*Pi^2. - Jaume Oliver Lafont, Jul 10 2017

Sum_{n>=4} (-1)^n/a(n) = 134 - 192*log(2). - Amiram Eldar, Oct 19 2020

a(n) = A000332(n) + 5*A000579(n+1). - Yasser Arath Chavez Reyes, Aug 18 2024

Extensions

Zabroki and Lajos formulas offset corrected by Gary Detlefs, Dec 05 2011