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%I #42 Sep 08 2022 08:45:19
%S 1,21,196,1176,5292,19404,60984,169884,429429,1002001,2186184,4504864,
%T 8836464,16604784,30046752,52581816,89311761,147685461,238369516,
%U 376372920,582481900,885069900,1322357400,1945206900,2820550005,4035556161,5702666256,7965629056
%N a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400.
%C Kekulé numbers for certain benzenoids.
%C 6th column of the table of Narayana numbers A001263. - _Zerinvary Lajos_, Jun 18 2007
%C 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 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 T. Koshy, Catalan Numbers with Applications, Oxford University Press, 2009, p. 7.
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=7. - _N. J. A. Sloane_, Aug 28 2010
%H T. D. Noe, <a href="/A108679/b108679.txt">Table of n, a(n) for n = 0..1000</a>
%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.
%F a(n) = A001263(n+6,6) = binomial(n+5, 5)*binomial(n+6, 5)/6 = binomial(n+6,6)*binomial(n+6,5)/(n+6).
%F 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 a(n) = binomial(n+5,5)^2 - binomial(n+5,4)*binomial(n+5,6). - _Gary Detlefs_, Dec 05 2011
%F a(n) = Product_{i=1..5} A002378(n+i)/A002378(i). - _Bruno Berselli_, Sep 01 2016
%F From _Amiram Eldar_, Oct 19 2020: (Start)
%F Sum_{n>=0} 1/a(n) = 27637/2 - 1400*Pi^2.
%F Sum_{n>=0} (-1)^n/a(n) = 2560*log(2) - 3547/2. (End)
%p a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400: seq(a(n),n=0..30);
%t 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 (Magma) [Binomial(n-1,5)*Binomial(n,5)/6: n in [6..35]]; // _Vincenzo Librandi_, Sep 04 2014
%o (PARI) Vec((1+10*x+20*x^2+10*x^3+x^4)/(1-x)^11 + O(x^99)) \\ _Altug Alkan_, Sep 02 2016
%Y Cf. A002378, A006542, A006857 (sequences having a similar structure).
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 17 2005
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