A108679 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400.

1, 21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, 582481900, 885069900, 1322357400, 1945206900, 2820550005, 4035556161, 5702666256, 7965629056

Offset: 0

Author

Emeric Deutsch, Jun 17 2005

Keywords

• nonn
• easy

Comments

Kekulé numbers for certain benzenoids.

6th column of the table of Narayana numbers A001263. - Zerinvary Lajos, Jun 18 2007

Sequence provided by binomial(n-1,m)*binomial(n,m)/(m+1) for m=5 and n>5 (these numbers are also called Runyon numbers, see T. Koshy in References). - Vincenzo Librandi, Sep 04 2014

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 1).
• T. Koshy, Catalan Numbers with Applications, Oxford University Press, 2009, p. 7.
• S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=7. - N. J. A. Sloane, Aug 28 2010

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.
• 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.

Crossrefs

Cf. A001263, A002378, A005585, A006542, A006857 (sequences having a similar structure), A288876.

Programs

Magma

[Binomial(n-1,5)*Binomial(n,5)/6: n in [6..45]]; // Vincenzo Librandi, Sep 04 2014

Maple

a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400: seq(a(n),n=0..40);

Mathematica

Table[(n + 1) (n + 2)^2 (n + 3)^2 (n + 4)^2 (n + 5)^2 (n + 6)/86400,{n, 0, 50}]  (* Harvey P. Dale, Mar 13 2011 *)

PARI

Vec((1+10*x+20*x^2+10*x^3+x^4)/(1-x)^11 + O(x^99)) \\ Altug Alkan, Sep 02 2016

SageMath

def A108679(n): return binomial(n+5,5)*binomial(n+6,5)//6
print([A108679(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

Formula

a(n) = binomial(n+5, 5)*binomial(n+6, 5)/6 = binomial(n+6, 6)*binomial(n+6, 5)/(n+6).

a(n) = A001263(n+6,6).

G.f.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1 - x)^11. Numerator polynomial is the fifth row polynomial of the Narayana triangle.

a(n) = binomial(n+5,5)^2 - binomial(n+5,4)*binomial(n+5,6). - Gary Detlefs, Dec 05 2011

a(n) = Product_{i=1..5} A002378(n+i)/A002378(i). - Bruno Berselli, Sep 01 2016

From Amiram Eldar, Oct 19 2020: (Start)

Sum_{n>=0} 1/a(n) = 27637/2 - 1400*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 2560*log(2) - 3547/2. (End)

a(n) = (A005585(n+2)^2 - A288876(n+1))/24. - Yasser Arath Chavez Reyes, Aug 19 2024