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A074082 Coefficient of q^2 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)=(1,1). 19

%I

%S 0,0,0,0,2,6,16,37,81,169,342,675,1307,2491,4686,8718,16066,29364,

%T 53282,96065,172215,307151,545286,963993,1697701,2979383,5211852,

%U 9090060,15810530,27429426,47473828,81983773,141286221,243011173

%N Coefficient of q^2 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)=(1,1).

%C The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).

%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_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).

%F G.f.: (2x^4-2x^6-x^7)/(1-x-x^2)^3.

%F a(n)=3a(n-1)-5a(n-3)+3a(n-5)+a(n-6) for n>=8.

%e The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^2 are 0,0,0,0,2,6.

%t b=1; lambda=1; expon=2; 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, 0}, LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 6, 16, 37}, 32]] (* _Jean-François Alcover_, Sep 23 2017 *)

%Y Coefficients of q^0, q^1 and q^3 are in A000045, A023610 and A074083. Related sequences with different values of b and lambda are in A074084-A074089.

%K nonn,easy

%O 0,5

%A Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

%E Edited by _Dean Hickerson_, Aug 21 2002

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Last modified April 20 03:02 EDT 2019. Contains 322293 sequences. (Running on oeis4.)