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A074085 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) = (2,1).

%I A074085 #21 Mar 03 2024 11:36:11
%S A074085 0,0,0,0,5,24,91,308,978,2978,8802,25440,72251,202316,559941,1534548,
%T A074085 4170256,11250630,30158900,80389600,213204513,562896832,1480086111,
%U A074085 3877337556,10123000126,26347306474,68378847990,176994780672
%N A074085 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) = (2,1).
%C A074085 The coefficient of q^0 is the Pell number A000129(n+1).
%H A074085 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 A074085 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, -9, -4, 9, 6, 1).
%F A074085 G.f.: (5*x^4 - 6*x^5 - 8*x^6 - 2*x^7)/(1 - 2*x - x^2)^3.
%F A074085 a(n) = 6*a(n-1) - 9*a(n-2) - 4*a(n-3) + 9*a(n-4) + 6*a(n-5) + a(n-6) for n >= 8.
%e A074085 The first 6 nu polynomials are nu(0) = 1, nu(1) = 2, nu(2) = 5, nu(3) = 12 + 2*q, nu(4) = 29 + 9*q + 5*q^2, nu(5) = 70 + 32q + 24*q^2 + 14*q^3 + 2*q^4, so the coefficients of q^2 are 0,0,0,0,5,24.
%t A074085 b=2; 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 A074085 (* Second program: *)
%t A074085 Join[{0,0}, LinearRecurrence[{6, -9, -4, 9, 6, 1}, {0, 0, 5, 24, 91, 308}, 30]] (* _Jean-François Alcover_, Dec 13 2018 *)
%Y A074085 Coefficients of q^0, q^1 and q^3 are in A000129, A074084 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.
%K A074085 nonn
%O A074085 0,5
%A A074085 Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002
%E A074085 Edited by _Dean Hickerson_, Aug 21 2002