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
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.
FORMULA
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).
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
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)
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..30);
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..35]]; // 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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 17 2005
STATUS
approved