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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)=(2,2).
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%I #17 Nov 18 2017 02:06:31

%S 0,0,0,0,0,40,232,1072,4400,16864,61728,218496,753792,2547840,8468608,

%T 27755776,89886976,288101888,915089920,2883416064,9021001728,

%U 28042881024,86672025600,266472878080,815347462144,2483820617728

%N 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)=(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.

%F Conjecture: O.g.f: 8*x^5*(1+x)*(12*x^4+24*x^3-2*x^2-16*x+5)/(2*x^2+2*x-1)^4. - _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^1 are 0,0,0,0,0,40.

%p nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074360 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,3) ) ; end: for n from 0 to 30 do printf("%d,", A074360(n)) ; od ; # _R. J. Mathar_, Sep 20 2006

%t nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n - 1] + 2*Total[q^Range[0, n - 2]]*nu[n - 2] // Expand;

%t a[n_] := Coefficient[nu[n], q, 3];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Nov 18 2017 *)

%Y Coefficient of q^0, q^1 and q^2 are in A002605, A074358 and A074359. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

%K nonn

%O 0,6

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

%E More terms from _R. J. Mathar_, Sep 20 2006