%I #42 Sep 08 2022 08:45:32

%S 1,45,825,9075,70785,429429,2147145,9202050,34763300,118195220,

%T 367479684,1057896060,2848181700,7229999700,17420856420,40067969766,

%U 88385227425,187746398125,385374185625,766691800875,1482270815025,2791289197125,5130235085625,9219552907500

%N Ninth column (and diagonal) of Narayana triangle A001263.

%C See a comment under A134288 on the coincidence of column and diagonal sequences.

%C Kekulé numbers K(O(1,8,n)) for certain benzenoids (see the Cyvin-Gutman reference, p. 105, eq. (i)).

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.

%H T. D. Noe, <a href="/A134290/b134290.txt">Table of n, a(n) for n = 0..1000</a>

%H W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, <a href="https://projecteuclid.org/euclid.mjms/1575342085">Problem 84</a>, Missouri Journal of Mathematical Sciences, volume 8, #2, spring 1996, pages 97-102.

%F a(n) = A001263(n+9,9) = binomial(n+9,9)*binomial(n+9,8)/(n+9).

%F 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.

%F a(n) = Product_{i=1..8} 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) = 497925669/175 - 288288*Pi^2.

%F Sum_{n>=0} (-1)^n/a(n) = 580367/35 - 1680*Pi^2. (End)

%p a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8))^2*(n+9))/14631321600:

%p seq(a(n), n=0..23); # _Peter Luschny_, Sep 01 2016

%t Table[Binomial[n+9,9]*Binomial[n+8,7]/8, {n,0,25}] (* _G. C. Greubel_, Aug 28 2019 *)

%o (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

%o (PARI) vector(25, n, binomial(n+8,9)*binomial(n+7,7)/8) \\ _G. C. Greubel_, Aug 28 2019

%o (Magma) [Binomial(n+9,9)*Binomial(n+8,7)/8: n in [0..25]]; // _G. C. Greubel_, Aug 28 2019

%o (Sage) [binomial(n+9,9)*binomial(n+8,7)/8 for n in (0..25)] # _G. C. Greubel_, Aug 28 2019

%o (GAP) List([0..25], n-> Binomial(n+9,9)*Binomial(n+8,7)/8); # _G. C. Greubel_, Aug 28 2019

%Y Cf. A002378.

%Y Cf. A134289 (eighth column of Narayana triangle).

%Y Cf. A134291 (tenth column of Narayana triangle).

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 13 2007