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A272379 a(n) = n*(24*n^3 - 60*n^2 + 54*n - 17).

%I A272379 #26 Sep 08 2022 08:46:16
%S A272379 0,1,86,759,3100,8765,19986,39571,70904,117945,185230,277871,401556,
%T A272379 562549,767690,1024395,1340656,1725041,2186694,2735335,3381260,
%U A272379 4135341,5009026,6014339,7163880,8470825,9948926,11612511,13476484,15556325,17868090,20428411
%N A272379 a(n) = n*(24*n^3 - 60*n^2 + 54*n - 17).
%H A272379 Vincenzo Librandi, <a href="/A272379/b272379.txt">Table of n, a(n) for n = 0..1000</a>
%H A272379 Richard P. Brent, <a href="http://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014 (page 16).
%H A272379 Richard P. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalising Tuenter's binomial sums</a>, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
%H A272379 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)
%F A272379 G.f.: x*(1 + 81*x + 339*x^2 + 155*x^3)/(1 - x)^5.
%F A272379 E.g.f.: x*(1 + 42*x + 84*x^2 + 24*x^3)*exp(x).
%F A272379 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A272379 See page 7 in Brent's paper:
%F A272379 a(n) = n^2*A272378(n) - n*(n-1)*A272378(n-1),
%F A272379 A272380(n) = n^2*a(n) - n*(n-1)*a(n-1).
%F A272379 From _Peter Bala_, Jan 30 2019: (Start)
%F A272379 Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have x^2*(24*x^3 - 60*x^2 + 54*x - 17) = Sum_{n >= 0} ((n+1)^9 + n^9)*a(n,x) and x*(24*x^3 - 60*x^2 + 54*x - 17)  = Sum_{n >= 0} ((n+1)^8 - n^8)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 9/2. See the Bala link in A036970. Cf. A272378 and A272380. (End)
%t A272379 Table[n (24 n^3 - 60 n^2 + 54 n - 17), {n, 0, 50}]
%t A272379 LinearRecurrence[{5,-10,10,-5,1},{0,1,86,759,3100},40] (* _Harvey P. Dale_, Mar 24 2021 *)
%o A272379 (Magma) [n*(24*n^3 - 60*n^2 + 54*n - 17): n in [0..50]];
%o A272379 (PARI) vector(100, n, n--; n*(24*n^3 - 60*n^2 + 54*n - 17)) \\ _Altug Alkan_, Apr 29 2016
%Y A272379 Cf. A272378, A272380.
%K A272379 nonn,easy
%O A272379 0,3
%A A272379 _Vincenzo Librandi_, Apr 29 2016