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%I #35 Sep 08 2022 08:45:32
%S 1,55,1210,15730,143143,1002001,5725720,27810640,118195220,449141836,
%T 1551580888,4936848280,14620666060,40648664980,106847919376,
%U 267119798440,638337753625,1464421905375,3237143159250,6917263803450
%N Tenth 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(1,9,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 G. C. Greubel, <a href="/A134291/b134291.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A001263(n+10,10) = binomial(n+10,10)*binomial(n+10,9)/(n+10).
%F O.g.f.: P(9,x)/(1-x)^19 with the numerator polynomial P(9,x) = Sum_{k=1..9} A001263(9,k)*x^(k-1), the ninth row polynomial of the Narayana triangle: P(9,x) = 1 + 36*x + 336*x^2 + 1176*x^3 + 1764*x^4 + 1176*x^5 + 336*x^6 + 36*x^7 + x^8.
%F a(n) = Product_{i=1..9} 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) = 2987553139/196 - 1544400*Pi^2.
%F Sum_{n>=0} (-1)^n/a(n) = 1179648*log(2)/7 - 114472793/980. (End)
%p a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9))^2*(n+10))/ 1316818944000:
%p seq(a(n), n=0..19); # _Peter Luschny_, Sep 01 2016
%t Table[Binomial[n + 10, 10]*Binomial[n + 10, 9]/(n + 10), {n, 0, 30}] (* _Wesley Ivan Hurt_, Apr 25 2017 *)
%o (PARI) vector(30, n, binomial(n+9,10)*binomial(n+8,8)/9) \\ _G. C. Greubel_, Aug 28 2019
%o (Magma) [Binomial(n+10,10)*Binomial(n+9,8)/9: n in [0..30]]; // _G. C. Greubel_, Aug 28 2019
%o (Sage) [binomial(n+10,10)*binomial(n+9,8)/9 for n in (0..30)] # _G. C. Greubel_, Aug 28 2019
%o (GAP) List([0..30], n-> Binomial(n+10,10)*Binomial(n+9,8)/9); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A001263, A002378, A134288.
%Y Cf. A134290 (ninth column of Narayana triangle).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 13 2007
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