A006542 - OEIS

%I M4707 #129 Jan 07 2025 12:57:32

%S 1,10,50,175,490,1176,2520,4950,9075,15730,26026,41405,63700,95200,

%T 138720,197676,276165,379050,512050,681835,896126,1163800,1495000,

%U 1901250,2395575,2992626,3708810,4562425,5573800,6765440,8162176,9791320,11682825,13869450

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

%C Number of permutations of n+4 that avoid the pattern 132 and have exactly 3 descents. - _Mike Zabrocki_, Aug 26 2004

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 20 2005

%C a(n) = number of Dyck n-paths with exactly 4 peaks. - _David Callan_, Jul 03 2006

%C 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

%D 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).

%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006542/b006542.txt">Table of n, a(n) for n = 4..200</a>

%H Isaac Ahern and Sam Cook, <a href="https://doi.org/10.33697/ajur.2016.024">Affine Symmetry Tensors in Minkowski Space</a>, American Journal of Undergraduate Research, Volume 13, Issue 3, August 2016.

%H P. Aluffi, <a href="http://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, 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]."]

%H Brandy Amanda Barnette, <a href="http://digitalcommons.wku.edu/theses/1484">Counting Convex Sets on Products of Totally Ordered Sets</a>, Masters Theses & Specialist Projects, Paper 1484, 2015.

%H V. E. Hoggatt, Jr., <a href="/A006542/a006542.pdf">Letter to N. J. A. Sloane, Apr 1977</a>

%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1967__10__23_0">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.

%H G. Kreweras, <a href="/A006542/a006542_1.pdf">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]

%H Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 6, 25.

%H Robert Munafo, <a href="http://mrob.com/pub/math/seq-a006542.html">C(n,3)C(n-1,3)/4</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

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

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

%F E.g.f.: x^4*(6 + 6*x + x^2)*exp(x)/144. - _Vladeta Jovovic_, Jan 29 2003

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

%F a(n) = C(n, 6) + 3*C(n+1, 6) + C(n+2, 6). - _Mike Zabrocki_, Aug 26 2004

%F G.f.: x^4*(1 + 3*x + x^2)/(1-x)^7. - _Emeric Deutsch_, Jun 20 2005

%F a(n) = C(n-2, n-4)*C(n-1, n-3)*C(n, n-2)/18. - _Zerinvary Lajos_, Jul 29 2005

%F a(n) = C(n,4)*C(n,3)/n. - _Mitch Harris_, Jul 06 2006

%F 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

%F a(n) = C(n-1,3)^2 - C(n-1,2)*C(n-1,4). - _Gary Detlefs_, Dec 05 2011

%F a(n) = A000292(A000217(n-1)) - A000217(A000292(n-1)). - _Ivan N. Ianakiev_, Jun 17 2014

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

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

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

%F a(n) = A000332(n) + 5*A000579(n+1). - _Yasser Arath Chavez Reyes_, Aug 18 2024

%p A006542:=-(1+3*z+z**2)/(z-1)**7; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p A006542:=n->n*((n-1)*(n-2))^2*(n-3)/144; seq(A006542(n), n=4..40); # _Wesley Ivan Hurt_, Jun 17 2014

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

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

%o (Magma) [ n*((n-1)*(n-2))^2*(n-3)/144 : n in [4..40] ]; // _Wesley Ivan Hurt_, Jun 17 2014

%o (Sage) [n*(n-1)^2*(n-2)^2*(n-3)/144 for n in (4..40)] # _G. C. Greubel_, Feb 24 2019

%o (GAP) List([4..40], n-> n*(n-1)^2*(n-2)^2*(n-3)/144); # _G. C. Greubel_, Feb 24 2019

%Y 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.

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

%Y Fourth column of the table of Narayana numbers A001263.

%Y Apart from a scale factor, a column of A124428.

%K nonn,easy,changed

%O 4,2

%A _N. J. A. Sloane_

%E Zabroki and Lajos formulas offset corrected by _Gary Detlefs_, Dec 05 2011