OFFSET
0,2
COMMENTS
See a comment under A134288 on the coincidence of column and diagonal sequences.
Kekulé numbers K(O(1,8,n)) for certain benzenoids (see the Cyvin-Gutman reference, p. 105, eq. (i)).
REFERENCES
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, Problem 84, Missouri Journal of Mathematical Sciences, volume 8, #2, spring 1996, pages 97-102.
FORMULA
a(n) = A001263(n+9,9) = binomial(n+9,9)*binomial(n+9,8)/(n+9).
O.g.f.: P(8,x)/(1-x)^17 with the numerator polynomial P(8,x) = Sum_{k=1..8} A001263(8,k)*x^(k-1), the eighth row polynomial of the Narayana triangle: P(8,x) = 1 + 28*x + 196*x^2 + 490*x^3 + 490*x^4 + 196*x^5 + 28*x^6 + x^7.
From Amiram Eldar, Oct 19 2020: (Start)
Sum_{n>=0} 1/a(n) = 497925669/175 - 288288*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 580367/35 - 1680*Pi^2. (End)
MAPLE
a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8))^2*(n+9))/14631321600:
seq(a(n), n=0..23); # Peter Luschny, Sep 01 2016
MATHEMATICA
Table[Binomial[n+9, 9]*Binomial[n+8, 7]/8, {n, 0, 25}] (* G. C. Greubel, Aug 28 2019 *)
PROG
(PARI) Vec((1+28*x+196*x^2+490*x^3+490*x^4+196*x^5+28*x^6+x^7)/(1-x)^17 + O(x^25)) \\ Altug Alkan, Sep 01 2016
(PARI) vector(25, n, binomial(n+8, 9)*binomial(n+7, 7)/8) \\ G. C. Greubel, Aug 28 2019
(Magma) [Binomial(n+9, 9)*Binomial(n+8, 7)/8: n in [0..25]]; // G. C. Greubel, Aug 28 2019
(Sage) [binomial(n+9, 9)*binomial(n+8, 7)/8 for n in (0..25)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..25], n-> Binomial(n+9, 9)*Binomial(n+8, 7)/8); # G. C. Greubel, Aug 28 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 13 2007
STATUS
approved