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%I #47 Sep 08 2022 08:45:00
%S 5,35,126,330,715,1365,2380,3876,5985,8855,12650,17550,23751,31465,
%T 40920,52360,66045,82251,101270,123410,148995,178365,211876,249900,
%U 292825,341055,395010,455126,521855,595665,677040,766480
%N Binomial coefficients binomial(2*n-3,4).
%C Number of intersections of diagonals in the interior of regular (2n-3)-gon. - _Philippe Deléham_, Jun 07 2013
%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="/A053126/b053126.txt">Table of n, a(n) for n = 4..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/wp-content/uploads/2017/10/enumfor.pdf">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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = binomial(2*n-3, 4) if n >= 4 else 0;
%F G.f.: (5+10*x+x^2)/(1-x)^5.
%F a(n) = A053123(n,4), n >= 4; a(n) = 0, n=0..3 (fifth column of shifted Chebyshev's S-triangle, decreasing order).
%F a(n) = A006561(2n-3). - _Philippe Deléham_, Jun 07 2013
%F E.g.f.: (90 - 84*x + 39*x^2 - 12*x^3 + 4*x^4)*exp(x)/6. - _G. C. Greubel_, Aug 26 2018
%F From _Amiram Eldar_, Jan 04 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 34/3 - 16*log(2).
%F Sum_{n>=4} (-1)^n/a(n) = 2*Pi - 4*log(2) - 10/3. (End)
%t Table[Binomial[2*n-3,4], {n,4,50}] (* _G. C. Greubel_, Aug 26 2018 *)
%o (Magma) [Binomial(2*n-3,4): n in [4..40]]; // _Vincenzo Librandi_, Oct 07 2011
%o (PARI) for(n=4,50, print1(binomial(2*n-3,4), ", ")) \\ _G. C. Greubel_, Aug 26 2018
%Y Cf. A053123, A002492.
%K nonn,easy
%O 4,1
%A _Wolfdieter Lang_
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