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

%I A108679 #60 Mar 13 2025 09:55:35
%S A108679 1,21,196,1176,5292,19404,60984,169884,429429,1002001,2186184,4504864,
%T A108679 8836464,16604784,30046752,52581816,89311761,147685461,238369516,
%U A108679 376372920,582481900,885069900,1322357400,1945206900,2820550005,4035556161,5702666256,7965629056
%N A108679 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400.
%C A108679 Kekulé numbers for certain benzenoids.
%C A108679 6th column of the table of Narayana numbers A001263. - _Zerinvary Lajos_, Jun 18 2007
%C A108679 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
%D A108679 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 1).
%D A108679 T. Koshy, Catalan Numbers with Applications, Oxford University Press, 2009, p. 7.
%D A108679 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=7. - _N. J. A. Sloane_, Aug 28 2010
%H A108679 T. D. Noe, <a href="/A108679/b108679.txt">Table of n, a(n) for n = 0..1000</a>
%H A108679 Brandy Amanda Barnette, <a href="https://digitalcommons.wku.edu/theses/1484">Counting Convex Sets on Products of Totally Ordered Sets</a>, Masters Theses & Specialist Projects, Paper 1484, 2015.
%H A108679 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 p. 25.
%F A108679 a(n) = binomial(n+5, 5)*binomial(n+6, 5)/6 = binomial(n+6, 6)*binomial(n+6, 5)/(n+6).
%F A108679 a(n) = A001263(n+6,6).
%F A108679 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.
%F A108679 a(n) = binomial(n+5,5)^2 - binomial(n+5,4)*binomial(n+5,6). - _Gary Detlefs_, Dec 05 2011
%F A108679 a(n) = Product_{i=1..5} A002378(n+i)/A002378(i). - _Bruno Berselli_, Sep 01 2016
%F A108679 From _Amiram Eldar_, Oct 19 2020: (Start)
%F A108679 Sum_{n>=0} 1/a(n) = 27637/2 - 1400*Pi^2.
%F A108679 Sum_{n>=0} (-1)^n/a(n) = 2560*log(2) - 3547/2. (End)
%F A108679 a(n) = (A005585(n+2)^2 - A288876(n+1))/24. - _Yasser Arath Chavez Reyes_, Aug 19 2024
%p A108679 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);
%t A108679 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 *)
%o A108679 (Magma) [Binomial(n-1,5)*Binomial(n,5)/6: n in [6..45]]; // _Vincenzo Librandi_, Sep 04 2014
%o A108679 (PARI) Vec((1+10*x+20*x^2+10*x^3+x^4)/(1-x)^11 + O(x^99)) \\ _Altug Alkan_, Sep 02 2016
%o A108679 (SageMath)
%o A108679 def A108679(n): return binomial(n+5,5)*binomial(n+6,5)//6
%o A108679 print([A108679(n) for n in range(41)]) # _G. C. Greubel_, Mar 12 2025
%Y A108679 Cf. A001263, A002378, A005585, A006542, A006857 (sequences having a similar structure), A288876.
%K A108679 nonn,easy
%O A108679 0,2
%A A108679 _Emeric Deutsch_, Jun 17 2005