Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #22 Sep 08 2022 08:46:16
%S 1,183,1205,3787,8649,16511,28093,44115,65297,92359,126021,167003,
%T 216025,273807,341069,418531,506913,606935,719317,844779,984041,
%U 1137823,1306845,1491827,1693489,1912551,2149733,2405755,2681337,2977199,3294061,3632643,3993665
%N a(n) = 120*n^3 + 60*n^2 + 2*n + 1.
%C 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 Vincenzo Librandi, <a href="/A272126/b272126.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F O.g.f.: (1 + 179*x + 479*x^2 + 61*x^3)/(1-x)^4.
%F E.g.f.: (1 + 182*x + 420*x^2 + 120*x^3)*exp(x).
%F a(n) = (2*n+1)*(60*n^2+1).
%F a(n) = (2*n+1) * A158673(n).
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
%F See page 7 in Brent's paper:
%F a(n) = (2*n+1)^2*A014641(n) - 2*n*(2*n+1)*A014641(n-1).
%F A272127(n) = (2*n+1)^2*a(n) - 2*n*(2*n+1)*a(n-1).
%F From _Peter Bala_, Jan 22 2019: (Start)
%F a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^6 * binomial(2*n + 1, n - k).
%F 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 Table[120 n^3 + 60 n^2 + 2 n + 1, {n, 0, 40}]
%t LinearRecurrence[{4,-6,4,-1},{1,183,1205,3787},40] (* _Harvey P. Dale_, Nov 08 2020 *)
%o (Magma) [120*n^3 + 60*n^2 + 2*n + 1: n in [0..50]];
%o (PARI) a(n) = 120*n^3 + 60*n^2 + 2*n + 1; \\ _Altug Alkan_, Apr 30 2016
%Y Cf. A014641, A158673, A160485, A245244, A272127, A272129.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Apr 25 2016