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A272126 a(n) = 120*n^3 + 60*n^2 + 2*n + 1.

%I A272126 #22 Sep 08 2022 08:46:16
%S A272126 1,183,1205,3787,8649,16511,28093,44115,65297,92359,126021,167003,
%T A272126 216025,273807,341069,418531,506913,606935,719317,844779,984041,
%U A272126 1137823,1306845,1491827,1693489,1912551,2149733,2405755,2681337,2977199,3294061,3632643,3993665
%N A272126 a(n) = 120*n^3 + 60*n^2 + 2*n + 1.
%C A272126 This is the polynomial Qbar(3,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials. - _Peter Bala_, Jan 22 2019
%H A272126 Vincenzo Librandi, <a href="/A272126/b272126.txt">Table of n, a(n) for n = 0..1000</a>
%H A272126 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 A272126 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 A272126 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A272126 O.g.f.: (1 + 179*x + 479*x^2 + 61*x^3)/(1-x)^4.
%F A272126 E.g.f.: (1 + 182*x + 420*x^2 + 120*x^3)*exp(x).
%F A272126 a(n) = (2*n+1)*(60*n^2+1).
%F A272126 a(n) = (2*n+1) * A158673(n).
%F A272126 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
%F A272126 See page 7 in Brent's paper:
%F A272126 a(n) = (2*n+1)^2*A014641(n) - 2*n*(2*n+1)*A014641(n-1).
%F A272126 A272127(n) = (2*n+1)^2*a(n) - 2*n*(2*n+1)*a(n-1).
%F A272126 From _Peter Bala_, Jan 22 2019: (Start)
%F A272126 a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^6 * binomial(2*n + 1, n - k).
%F A272126 a(n-1) = 2/4^n * binomial(2*n,n) * ( 1 + 3^6*(n - 1)/(n + 1) + 5^6*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^6*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)
%t A272126 Table[120 n^3 + 60 n^2 + 2 n + 1, {n, 0, 40}]
%t A272126 LinearRecurrence[{4,-6,4,-1},{1,183,1205,3787},40] (* _Harvey P. Dale_, Nov 08 2020 *)
%o A272126 (Magma) [120*n^3 + 60*n^2 + 2*n + 1: n in [0..50]];
%o A272126 (PARI) a(n) = 120*n^3 + 60*n^2 + 2*n + 1; \\ _Altug Alkan_, Apr 30 2016
%Y A272126 Cf. A014641, A158673, A160485, A245244, A272127, A272129.
%K A272126 nonn,easy
%O A272126 0,2
%A A272126 _Vincenzo Librandi_, Apr 25 2016