# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a134290 Showing 1-1 of 1 %I A134290 #42 Sep 08 2022 08:45:32 %S A134290 1,45,825,9075,70785,429429,2147145,9202050,34763300,118195220, %T A134290 367479684,1057896060,2848181700,7229999700,17420856420,40067969766, %U A134290 88385227425,187746398125,385374185625,766691800875,1482270815025,2791289197125,5130235085625,9219552907500 %N A134290 Ninth column (and diagonal) of Narayana triangle A001263. %C A134290 See a comment under A134288 on the coincidence of column and diagonal sequences. %C A134290 Kekulé numbers K(O(1,8,n)) for certain benzenoids (see the Cyvin-Gutman reference, p. 105, eq. (i)). %D A134290 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988. %H A134290 T. D. Noe, Table of n, a(n) for n = 0..1000 %H A134290 W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, Problem 84, Missouri Journal of Mathematical Sciences, volume 8, #2, spring 1996, pages 97-102. %F A134290 a(n) = A001263(n+9,9) = binomial(n+9,9)*binomial(n+9,8)/(n+9). %F A134290 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 A134290 a(n) = Product_{i=1..8} A002378(n+i)/A002378(i). - _Bruno Berselli_, Sep 01 2016 %F A134290 From _Amiram Eldar_, Oct 19 2020: (Start) %F A134290 Sum_{n>=0} 1/a(n) = 497925669/175 - 288288*Pi^2. %F A134290 Sum_{n>=0} (-1)^n/a(n) = 580367/35 - 1680*Pi^2. (End) %p A134290 a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8))^2*(n+9))/14631321600: %p A134290 seq(a(n), n=0..23); # _Peter Luschny_, Sep 01 2016 %t A134290 Table[Binomial[n+9,9]*Binomial[n+8,7]/8, {n,0,25}] (* _G. C. Greubel_, Aug 28 2019 *) %o A134290 (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 A134290 (PARI) vector(25, n, binomial(n+8,9)*binomial(n+7,7)/8) \\ _G. C. Greubel_, Aug 28 2019 %o A134290 (Magma) [Binomial(n+9,9)*Binomial(n+8,7)/8: n in [0..25]]; // _G. C. Greubel_, Aug 28 2019 %o A134290 (Sage) [binomial(n+9,9)*binomial(n+8,7)/8 for n in (0..25)] # _G. C. Greubel_, Aug 28 2019 %o A134290 (GAP) List([0..25], n-> Binomial(n+9,9)*Binomial(n+8,7)/8); # _G. C. Greubel_, Aug 28 2019 %Y A134290 Cf. A002378. %Y A134290 Cf. A134289 (eighth column of Narayana triangle). %Y A134290 Cf. A134291 (tenth column of Narayana triangle). %K A134290 nonn,easy %O A134290 0,2 %A A134290 _Wolfdieter Lang_, Nov 13 2007 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE