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A272378 a(n) = n*(6*n^2 - 8*n + 3).

%I A272378 #35 Sep 08 2022 08:46:16
%S A272378 0,1,22,99,268,565,1026,1687,2584,3753,5230,7051,9252,11869,14938,
%T A272378 18495,22576,27217,32454,38323,44860,52101,60082,68839,78408,88825,
%U A272378 100126,112347,125524,139693,154890,171151,188512,207009,226678,247555,269676,293077
%N A272378 a(n) = n*(6*n^2 - 8*n + 3).
%H A272378 Vincenzo Librandi, <a href="/A272378/b272378.txt">Table of n, a(n) for n = 0..1000</a>
%H A272378 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 A272378 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 A272378 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1)
%F A272378 G.f.: x*(1 + 18*x + 17*x^2)/(1 - x)^4.
%F A272378 E.g.f.: x*(1 + 10*x + 6*x^2)*exp(x).
%F A272378 a(n) = n*A080859(n+1).
%F A272378 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3.
%F A272378 See page 7 in Brent's paper:
%F A272378 a(n) = n^2*A000384(n) - n*(n-1)*A000384(n-1).
%F A272378 A272379(n) = n^2*a(n) - n*(n-1)*a(n-1).
%F A272378 From _Peter Bala_, Jan 30 2019: (Start)
%F A272378 Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have x^2*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^7 + n^7)*a(n,x) and x*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^6 - n^6)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 7/2. See the Bala link in A036970. Cf. A272379. (End)
%t A272378 Table[n (6 n^2 - 8 n + 3), {n, 0, 50}]
%t A272378 LinearRecurrence[{4,-6,4,-1},{0,1,22,99},40] (* _Harvey P. Dale_, Dec 29 2017 *)
%o A272378 (Magma) [n*(6*n^2 - 8*n + 3): n in [0..50]];
%o A272378 (PARI) vector(100, n, n--; n*(6*n^2 - 8*n + 3)) \\ _Altug Alkan_, Apr 29 2016
%o A272378 (Python) for n in range(0,10**3):print(n*(6*n**2-8*n+3),end=", ") # _Soumil Mandal_, Apr 30 2016
%Y A272378 Cf. A000384, A015237, A272379, A272380.
%K A272378 nonn,easy
%O A272378 0,3
%A A272378 _Vincenzo Librandi_, Apr 29 2016