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A074363 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)=(3,1).

%I A074363 #22 Aug 02 2025 09:58:46
%S A074363 0,0,0,0,0,36,246,1293,6057,26592,111934,457353,1827529,7176636,
%T A074363 27789976,106371588,403204880,1515647250,5656172420,20974163475,
%U A074363 77339044883,283743384228,1036296662574,3769287797151,13658724680991,49325767966842,177570110818794
%N A074363 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)=(3,1).
%C A074363 Coefficient of q^0 is A006190(n+1).
%H A074363 Colin Barker, <a href="/A074363/b074363.txt">Table of n, a(n) for n = 0..1001</a>
%H A074363 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 A074363 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (12,-50,72,21,-72,-50,-12,-1).
%F A074363 From _Colin Barker_, Nov 18 2017: (Start)
%F A074363 G.f.: x^4*(3 + x)*(12 - 66*x + 69*x^2 + 60*x^3 + 10*x^4) / (1 - 3*x - x^2)^4.
%F A074363 a(n) = 12*a(n-1) - 50*a(n-2) + 72*a(n-3) + 21*a(n-4) - 72*a(n-5) - 50*a(n-6) - 12*a(n-7) - a(n-8) for n>9.
%F A074363 (End)
%e A074363 The first 6 nu polynomials are nu(0)=1, nu(1)=3, nu(2)=10, nu(3)=33+3q, nu(4)=109+19q+10q^2, nu(5)=360+93q+66q^2+36q^3+3q^4, so the coefficients of q^1 are 0,0,0,0,0,36.
%o A074363 (PARI) concat(vector(4), Vec(x^4*(3 + x)*(12 - 66*x + 69*x^2 + 60*x^3 + 10*x^4) / (1 - 3*x - x^2)^4 + O(x^40))) \\ _Colin Barker_, Nov 18 2017
%Y A074363 Coefficients of q^0, q^1 and q^2 are in A006190, A074361 and A074362. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074360.
%K A074363 nonn,easy
%O A074363 0,6
%A A074363 Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%E A074363 More terms from Brent Lehman (mailbjl(AT)yahoo.com), Aug 25 2002
%E A074363 Missing a(0)=0 inserted by _Sean A. Irvine_, Jan 20 2025
%E A074363 Missing a(0)=0 inserted in b-file by _David Radcliffe_, Aug 01 2025