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%I #36 Sep 08 2022 08:45:32
%S 1,28,336,2520,13860,60984,226512,736164,2147145,5725720,14158144,
%T 32821152,71954064,150233760,300467520,578399976,1075994073,
%U 1941008916,3405278800,5824819000,9735768900,15931258200,25565576400,40293571500,62455035825,95315993136
%N a(n) = binomial(n+7,7)*binomial(n+7,6)/(n+7).
%C Seventh column of Narayana triangle A001263.
%C In the Narayana triangle N(n,k)= A001263(n,k) the sequence of column nr. k>=1 (without leading zeros coincides with the sequence of the diagonal d=k-1>=0 (d=0 for the main diagonal N(n,n)).
%C Kekulé numbers K(O(1,6,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.
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; Prop. 8.4, case n=8. - _N. J. A. Sloane_, Aug 28 2010.
%H T. D. Noe, <a href="/A134288/b134288.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F a(n) = A001263(n+7,7).
%F O.g.f.: (1 + 15*x + 50*x^2 + 50*x^3 + 15*x^4 + x^5)/(1-x)^13. Numerator polynomial is the sixth row polynomial of the Narayana triangle.
%F a(n) = binomial(n+6,6)^2 - binomial(n+6,5)*binomial(n+6,7). - _Gary Detlefs_, Dec 05 2011
%F a(n) = Product_{i=1..6} 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) = 1741019/20 - 8820*Pi^2.
%F Sum_{n>=0} (-1)^n/a(n) = 210*Pi^2 - 41433/20. (End)
%p a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6))^2*(n+7))/3628800:
%p seq(a(n), n=0..25); # _Peter Luschny_, Sep 01 2016
%t Table[Binomial[n+7,7] Binomial[n+7,6]/(n+7),{n,0,30}] (* or *) LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13, 1}, {1,28,336,2520,13860,60984,226512,736164,2147145,5725720, 14158144, 32821152,71954064}, 30] (* _Harvey P. Dale_, Sep 28 2016 *)
%o (PARI) Vec((1+15*x+50*x^2+50*x^3+15*x^4+x^5)/(1-x)^13 + O(x^30)) \\ _Altug Alkan_, Sep 01 2016
%o (PARI) vector(30, n, binomial(n+6, 7)*binomial(n+5, 5)/6) \\ _G. C. Greubel_, Aug 27 2019
%o (Magma) [Binomial(n+7, 7)*Binomial(n+6, 5)/6: n in [0..30]]; // _G. C. Greubel_, Aug 27 2019
%o (Sage) [binomial(n+7, 7)*binomial(n+6, 5)/6 for n in (0..30)] # _G. C. Greubel_, Aug 27 2019
%o (GAP) List([0..30], n-> Binomial(n+7, 7)*Binomial(n+6, 5)/6); # _G. C. Greubel_, Aug 27 2019
%Y Cf. A002378.
%Y Cf. A108679 (sixth column of Narayana triangle).
%Y Cf. A134289 (eighth column of Narayana triangle).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 13 2007
%E Edited by _N. J. A. Sloane_, Aug 28 2010
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