login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = 120*n^3 + 60*n^2 + 2*n + 1.
4

%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