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URL: https://oeis.org/A108679

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A108679
a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400.
9
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
0,2
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
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.
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
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 *)
PROG
(Magma) [Binomial(n-1, 5)*Binomial(n, 5)/6: n in [6..45]]; // Vincenzo Librandi, Sep 04 2014
(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
CROSSREFS
Cf. A001263, A002378, A005585, A006542, A006857 (sequences having a similar structure), A288876.
Sequence in context: A163718 A164606 A027780 * A200825 A058086 A270957
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 17 2005
STATUS
approved