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%I #39 Oct 21 2022 09:45:20
%S 7,84,462,1716,5005,12376,27132,54264,100947,177100,296010,475020,
%T 736281,1107568,1623160,2324784,3262623,4496388,6096454,8145060,
%U 10737573,13983816,18009460,22957480,28989675,36288252,45057474,55525372,67945521,82598880,99795696
%N Binomial coefficients C(2*n-5,6).
%C a(n) = A053123(n,6), n >= 6; a(n) = 0, n=0..5, (seventh column of shifted Chebyshev's S-triangle, decreasing order).
%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="/A053128/b053128.txt">Table of n, a(n) for n = 6..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>, 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/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F a(n) = binomial(2*n-5, 6) if n >= 6 else 0.
%F G.f.: (7+35*x+21*x^2+x^3)/(1-x)^7.
%F E.g.f.: (18900 - 16380*x + 6975*x^2 - 1935*x^3 + 390*x^4 - 60*x^5 + 8*x^6)*exp(x)/90. - _G. C. Greubel_, Aug 26 2018
%F a(n) = (n-5)*(n-4)*(n-3)*(2*n-9)*(2*n-7)*(2*n-5)/90. - _Wesley Ivan Hurt_, Mar 25 2020
%F From _Amiram Eldar_, Oct 21 2022: (Start)
%F Sum_{n>=6} 1/a(n) = 667/10 - 96*log(2).
%F Sum_{n>=6} (-1)^n/a(n) = 273/10 - 6*Pi - 12*log(2). (End)
%t Table[Binomial[2*n-5,6], {n,6,50}] (* _G. C. Greubel_, Aug 26 2018 *)
%o (Magma) [Binomial(2*n-5,6): n in [6..40]]; // _Vincenzo Librandi_, Oct 07 2011
%o (PARI) for(n=6,50, print1(binomial(2*n-5,6), ", ")) \\ _G. C. Greubel_, Aug 26 2018
%Y Cf. A053123, A053127.
%K nonn,easy
%O 6,1
%A _Wolfdieter Lang_
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