A053131 - OEIS

%I #35 Oct 21 2022 09:45:07

%S 10,220,2002,11440,48620,167960,497420,1307504,3124550,6906900,

%T 14307150,28048800,52451256,94143280,163011640,273438880,445891810,

%U 708930508,1101716330,1677106640,2505433700,3679075400,5317936260,7575968400,10648873950,14783142660

%N Binomial coefficients C(2*n-8,9).

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

%H Vincenzo Librandi, <a href="/A053131/b053131.txt">Table of n, a(n) for n = 9..200</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Milan Janjić, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%F a(n) = binomial(2*n-8, 9) if n >= 9 else 0.

%F G.f.: (10+120*x+252*x^2+120*x^3+10*x^4)/(1-x)^10.

%F a(n) = 2*A053133(n).

%F a(n) = -A053123(n,9), n >= 9; a(n) := 0, n=0..8 (tenth column of shifted Chebyshev's S-triangle, decreasing order).

%F From _Amiram Eldar_, Oct 21 2022: (Start)

%F Sum_{n>=9} 1/a(n) = 223611/280 - 1152*log(2).

%F Sum_{n>=9} (-1)^(n+1)/a(n) = 72*log(2) - 13947/280. (End)

%t Binomial[2*Range[9,40]-8,9] (* _Harvey P. Dale_, Mar 19 2012 *)

%o (Magma)[Binomial(2*n-8,9): n in [9..40]]; // _Vincenzo Librandi_, Oct 07 2011

%o (PARI) for(n=9,50, print1(binomial(2*n-8,9), ", ")) \\ _G. C. Greubel_, Aug 26 2018

%Y Cf. A053123, A053130, A053133.

%K nonn,easy

%O 9,1

%A _Wolfdieter Lang_