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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)=(2,2).
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%I #18 Sep 23 2017 11:23:56

%S 0,0,0,0,12,64,280,1088,3968,13856,46912,155136,503616,1610496,

%T 5086336,15895552,49229312,151275008,461662208,1400356864,4224703488,

%U 12683452416,37911164928,112865394688,334788444160,989756825600

%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)=(2,2).

%C Coefficient of q^0 is A002605.

%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 (6, -6, -16, 12, 24, 8).

%F Conjecture: O.g.f: 4*x^4*(-3+2*x+8*x^2+4*x^3)/(2*x^2+2*x-1)^3. - _R. J. Mathar_, Jul 22 2009

%e The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^2 are 0,0,0,0,12,64.

%p nu := proc(n,b,lambda) if n = 0 then 1 ; elif n = 1 then b ; else b*nu(n-1,b,lambda)+lambda*nu(n-2,b,lambda)*add(q^i,i=0..n-2) ; fi ; end: A074359 := proc(n) local b,lambda,thisnu ; b := 2 ; lambda := 2 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074359(n) ) ; od ; # _R. J. Mathar_, Mar 20 2007

%t Join[{0, 0}, LinearRecurrence[{6, -6, -16, 12, 24, 8}, {0, 0, 12, 64, 280, 1088}, 24]] (* _Jean-François Alcover_, Sep 23 2017 *)

%Y Coefficient of q^0, q^1 and q^3 are in A002605, A074358 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

%K nonn

%O 0,5

%A Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

%E More terms from _R. J. Mathar_, Mar 20 2007