A006857 a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5).

1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, 26883780, 37823500, 52474500, 71867250, 97260345, 130179231, 172459665, 226296280, 294296640, 379541184

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy

Comments

Number of permutations of n+5 that avoid the pattern 132 and have exactly 4 descents.

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005

Partial sums of A114242. - Peter Bala, Sep 21 2007

Dimensions of certain Lie algebra (see reference for precise definition).

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/1).
• S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 239.
• 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 = 0..1000
• Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
• 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]
• J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.3, case a=4]
• 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 p. 25.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

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.

5th column of the table of Narayana numbers A001263.

Cf. A002378, A114242.

Programs

Magma

A006857:= func< n | Binomial(n+4,3)*Binomial(n+5,5)/4 >;
[A006857(n): n in [0..40]]; // G. C. Greubel, Mar 12 2025

Maple

a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)/2880: seq(a(n),n=0..38); # Emeric Deutsch, Nov 18 2005

Mathematica

Table[Binomial[n+5,5] * Binomial[n+5,4]/(n+5), {n, 0, 50}] (* T. D. Noe, May 29 2012 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,15,105,490,1764,5292,13860,32670,70785},40] (* Harvey P. Dale, Oct 19 2024 *)

PARI

a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5) \\ Charles R Greathouse IV, Jun 11 2015

PARI

Vec((1+6*x+6*x^2+x^3)/(1-x)^9 + O(x^99)) \\ Altug Alkan, Sep 01 2016

SageMath

def A006857(n): return binomial(n+4,3)*binomial(n+5,5)//4
print([A006857(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

Formula

From - Vladeta Jovovic, Jan 29 2003: (Start)

a(n) = (n+4)!*(n+5)!/(2880*n!*(n+1)!).

E.g.f.: 1/2880*(2880 + 40320*x + 109440*x^2 + 105120*x^3 + 45000*x^4 + 9504*x^5 + 1016*x^6 + 52*x^7 + x^8)*exp(x). (End)

From Mike Zabrocki, Aug 26 2004: (Start)

a(n) = C(n+5,8) + 6*C(n+6,8) + 6*C(n+7,8) + C(n+8,8).

a(n) = C(n+4,4)*C(n+5,4)/5.

O.g.f.: (1 + 6*x + 6*x^2 + x^3)/(1-x)^9. (End)

From Wolfdieter Lang, Nov 13 2007: (Start)

a(n) = A001263(n+5,5).

Numerator polynomial of the g.f is the fourth row polynomial of the Narayana triangle. (End)

a(n)= C(n+4,4)^2 - C(n+4,3)*C(n+4,5). - Gary Detlefs, Dec 05 2011

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

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 25 * (79 - 8*Pi^2).

Sum_{n>=0} (-1)^n/a(n) = 595/3 - 20*Pi^2. (End)

Extensions

More terms from Vladeta Jovovic, Jan 29 2003

Better description from Mike Zabrocki, Aug 26 2004

New definition from N. J. A. Sloane, Aug 28 2010

Zabrocki formulas offset corrected by Gary Detlefs, Dec 05 2011