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%I #25 Sep 08 2022 08:45:00
%S 3,28,126,396,1001,2184,4284,7752,13167,21252,32890,49140,71253,
%T 100688,139128,188496,250971,329004,425334,543004,685377,856152,
%U 1059380,1299480,1581255,1909908,2291058,2730756,3235501,3812256,4468464
%N One half of binomial coefficients C(2*n-4,5).
%H Vincenzo Librandi, <a href="/A053132/b053132.txt">Table of n, a(n) for n = 5..200</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)/2 if n >= 5 else 0.
%F G.f.: (x^5)*(3+10*x+3*x^2)/(1-x)^6.
%F a(n) = A053127(n)/2
%F a(n) = Sum_{k=1..n-4} (A000217(k)*A000217(2*n-k-7)). - _Reinhard Zumkeller_, Mar 03 2015
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=5} 1/a(n) = 335/6 - 80*log(2).
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 85/6 - 20*log(2). (End)
%t Binomial[2*Range[5,40]-4,5]/2 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3,28,126,396,1001,2184},40] (* _Harvey P. Dale_, Oct 25 2015 *)
%o (Magma) [Binomial(2*n-4,5)/2: n in [5..40]]; // _Vincenzo Librandi_, Oct 07 2011
%o (Haskell)
%o a053132 n = a053132_list !! (n-5)
%o a053132_list = f [1] $ drop 2 a000217_list where
%o f xs ts'@(t:ts) = (sum $ zipWith (*) xs ts') : f (t:xs) ts
%o -- _Reinhard Zumkeller_, Mar 03 2015
%o (PARI) for(n=5,50, print1(binomial(2*n-4,5)/2, ", ")) \\ _G. C. Greubel_, Aug 26 2018
%Y Cf. A000217, A053127.
%K nonn,easy
%O 5,1
%A _Wolfdieter Lang_
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