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 A134288 a(n) = binomial(n+7,7)*binomial(n+7,6)/(n+7). 8

%I

%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 KekulĂ© 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, KekulĂ© 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

%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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Last modified March 31 18:30 EDT 2020. Contains 333151 sequences. (Running on oeis4.)