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%I #43 Sep 08 2022 08:45:32
%S 1,36,540,4950,32670,169884,736164,2760615,9202050,27810640,77364144,
%T 200443464,488259720,1126753200,2478857040,5226256926,10606227291,
%U 20796524100,39525557500,73018266750,131432880150,231003243900,397179490500,669161098125,1106346348900
%N Eighth 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(1,7,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="/A134289/b134289.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
%F a(n) = A001263(n+8,8) = binomial(n+8,8)*binomial(n+8,7)/(n+8).
%F O.g.f.: P(7,x)/(1-x)^15 with the numerator polynomial P(7,x) = Sum_{k=1..7} A001263(7,k)*x^(k-1), the seventh row polynomial of the Narayana triangle: P(7,x) = 1 + 21*x + 105*x^2 + 175*x^3 + 105*x^4 + 21*x^5 + x^6.
%F For n>14: a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15). - _Harvey P. Dale_, Jul 23 2012
%F a(n) = Product_{i=1..7} 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) = 12767346/25 - 51744*Pi^2.
%F Sum_{n>=0} (-1)^n/a(n) = 1192508/75 - 114688*log(2)/5. (End)
%p a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7))^2*(n+8))/203212800;
%p seq(a(n), n=0..24); # _Peter Luschny_, Sep 01 2016
%t Table[(Binomial[n + 8, 8] Binomial[n + 8, 7])/(n + 8), {n, 0, 30}] (* or *) LinearRecurrence[{15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1},{1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050, 27810640, 77364144, 200443464, 488259720, 1126753200, 2478857040}, 30] (* _Harvey P. Dale_, Jul 23 2012 *)
%o (PARI) vector(30, n, binomial(n+7,8)*binomial(n+6,6)/7) \\ _G. C. Greubel_, Aug 28 2019
%o (Magma) [Binomial(n+8,8)*Binomial(n+7,6)/7: n in [0..30]]; // _G. C. Greubel_, Aug 28 2019
%o (Sage) [binomial(n+8,8)*binomial(n+7,6)/7 for n in (0..30)] # _G. C. Greubel_, Aug 28 2019
%o (GAP) List([0..30], n-> Binomial(n+8,8)*Binomial(n+7,6)/7); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A002378.
%Y Cf. A134288 (seventh column of Narayana triangle).
%Y Cf. A134290 (ninth column of Narayana triangle).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 13 2007
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