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%I #24 Sep 08 2022 08:45:06
%S 0,0,0,6,33,144,570,2118,7587,26448,90420,304470,1013061,3338112,
%T 10911150,35423862,114342855,367242336,1174368360,3741029094,
%U 11876859369,37591894320,118659631650,373630740966,1173847761003
%N Coefficient of q^1 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 The coefficient of q^0 is A014983(n+1).
%H G. C. Greubel, <a href="/A074087/b074087.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,-12,-9).
%F G.f.: (6*x^3 +9*x^4)/(1-2*x-3*x^2)^2.
%F a(n) = 4*a(n-1) +2*a(n-2) -12*a(n-3) -9*a(n-4) for n>=5.
%e 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^1 are 0,0,0,6,33,144.
%t b=2; lambda=3; expon=1; 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 (* Second program: *)
%t Join[{0}, LinearRecurrence[{4,2,-12,-9}, {0,0,6,33}, 50]] (* _G. C. Greubel_, May 26 2018 *)
%o (PARI) x='x+O('x^30); concat([0,0,0], Vec((6*x^3 +9*x^4)/(1-2*x-3*x^2)^2)) \\ _G. C. Greubel_, May 26 2018
%o (Magma) I:=[0,0,6,33]; [0] cat [n le 4 select I[n] else 4*Self(n-1) + 2*Self(n-2) -12*Self(n-3) -9*Self(n-4): n in [1..30]]; // _G. C. Greubel_, May 26 2018
%Y Coefficients of q^0, q^2 and q^3 are in A014983, A074088 and A074089. Related sequences with other values of b and lambda are in A074082-A074086.
%K nonn,easy
%O 0,4
%A Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002
%E Edited by _Dean Hickerson_, Aug 21 2002
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