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%I #46 Sep 08 2022 08:45:00
%S 6,56,252,792,2002,4368,8568,15504,26334,42504,65780,98280,142506,
%T 201376,278256,376992,501942,658008,850668,1086008,1370754,1712304,
%U 2118760,2598960,3162510,3819816,4582116,5461512,6471002,7624512
%N Binomial coefficients C(2*n-4,5).
%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="/A053127/b053127.txt">Table of n, a(n) for n = 5..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 Janjic, <a href="https://pmf.unibl.org/janjic/">Two Enumerative Functions</a>, University of Banja Luka (Bosnia and Herzegovina, 2017).
%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = binomial(2*n-4, 5) if n >= 5 else 0.
%F a(n) = -A053123(n,5), n >= 5; a(n) := 0, n=0..4 (sixth column of shifted Chebyshev's S-triangle, decreasing order).
%F G.f.: (6+20*x+6*x^2)/(1-x)^6.
%F a(n) = 6*a(n-1) - 15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). - _Harvey P. Dale_, Jun 03 2013
%F E.g.f.: (-840 + 750*x - 330*x^2 + 95*x^3 - 20*x^4 + 4*x^5)*exp(x)/15. - _G. C. Greubel_, Aug 26 2018
%F a(n) = (2*n-8)*(2*n-7)*(2*n-6)*(2*n-5)*(2*n-4)/120. - _Wesley Ivan Hurt_, Mar 25 2020
%F From _Amiram Eldar_, Jan 03 2022: (Start)
%F Sum_{n>=5} 1/a(n) = 335/12 - 40*log(2).
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 85/12 - 10*log(2). (End)
%t Binomial[2Range[5,40]-4,5] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{6,56,252,792,2002,4368},30] (* _Harvey P. Dale_, Jun 03 2013 *)
%o (Magma) [Binomial(2*n-4,5): n in [5..40]]; // _Vincenzo Librandi_, Oct 07 2011
%o (Haskell)
%o a053127 = (* 2) . a053132 -- _Reinhard Zumkeller_, Mar 03 2015
%o (PARI) for(n=5,50, print1(binomial(2*n-4,5), ", ")) \\ _G. C. Greubel_, Aug 26 2018
%Y Cf. A053123, A053132, A053126.
%K nonn,easy
%O 5,1
%A _Wolfdieter Lang_
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