This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A074089 #26 Sep 08 2022 08:45:06
%S A074089 0,0,0,0,0,78,501,2574,11757,50034,203229,797316,3046362,11394774,
%T A074089 41885913,151732722,542840175,1921208586,6735519249,23417342568,
%U A074089 80810560596,277008392478,943826398893,3198199361910,10783017814065
%N A074089 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = b*nu(n-1) + lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,3).
%C A074089 The coefficient of q^0 is A014983(n+1).
%H A074089 G. C. Greubel, <a href="/A074089/b074089.txt">Table of n, a(n) for n = 0..1000</a>
%H A074089 M. Beattie, S. Dăscălescu and S. Raianu, <a href="https://arxiv.org/abs/math/0204075">Lifting of Nichols Algebras of Type B_2</a>, arXiv:math/0204075 [math.QA], 2002.
%H A074089 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12,-40,74,120,-108,-216,-81).
%F A074089 G.f.: (78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4.
%F A074089 a(n) = 8*a(n-1) -12*a(n-2) -40*a(n-3) +74*a(n-4) +120*a(n-5) -108*a(n-6) -216*a(n-7) -81*a(n-8) for n>=11.
%e A074089 The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=7, nu(3) = 20 + 6q, nu(4) = 61 + 33q + 21q^2, nu(5) = 182 + 144q + 120q^2 + 78q^3 + 18q^4, so the coefficients of q^3 are 0,0,0,0,0,78.
%t A074089 b=2; lambda=3; expon=3; nu[0]=1; nu[1]=b; nu[n_] := nu[n]= Together[ b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
%t A074089 (* Second program: *)
%t A074089 CoefficientList[Series[(78*x^5-123*x^6-498*x^7+297*x^8+1134*x^9 + 567*x^10)/( 1-2*x-3*x^2)^4, {x, 0, 50}], x] (* _G. C. Greubel_, May 26 2018 *)
%o A074089 (PARI) x='x+O('x^30); concat([0,0,0,0,0], Vec((78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4)) \\ _G. C. Greubel_, May 26 2018
%o A074089 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0,0,0,0,0] cat Coefficients(R!((78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4)); // _G. C. Greubel_, May 26 2018
%Y A074089 Coefficients of q^0, q^1 and q^2 are in A014983, A074087 and A074088. Related sequences with other values of b and lambda are in A074082-A074086.
%K A074089 nonn,easy
%O A074089 0,6
%A A074089 Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002
%E A074089 Edited by _Dean Hickerson_, Aug 21 2002