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%I #23 Jan 27 2019 03:12:45
%S 0,0,0,2,9,32,102,306,883,2480,6828,18514,49597,131568,346194,904738,
%T 2350695,6076960,15641304,40103778,102473969,261046144,663180222,
%U 1680628946,4249496795,10722962256,27007159428,67904097074
%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)=(2,1).
%C The coefficient of q^0 is the Pell number A000129(n+1).
%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, -4, -1).
%F G.f.: (2x^3+x^4)/(1-2x-x^2)^2.
%F a(n) = 4a(n-1)-2a(n-2)-4a(n-3)-a(n-4) for n>=5.
%e The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^1 are 0,0,0,2,9,32.
%t b=2; lambda=1; 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,-4,-1},{0,0,2,9},30]] (* _Harvey P. Dale_, Apr 18 2012 *)
%Y Coefficients of q^0, q^2 and q^3 are in A000129, A074085 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.
%K nonn
%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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