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Coefficient of q^1 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 #26 Nov 18 2017 02:45:34

%S 0,0,0,4,20,80,288,976,3184,10112,31488,96576,292672,878336,2614784,

%T 7731456,22728448,66482176,193617920,561718272,1624101888,4681535488,

%U 13457924096,38592008192,110419341312,315287830528,898583560192

%N Coefficient of q^1 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 G.f.: 4*x^3*(x + 1)/(2*x^2 + 2*x - 1)^2 (conjectured). - _Chai Wah Wu_, May 30 2016

%F a(n) = (1/18)*((1 + sqrt(3))^n*(-9 + 2*sqrt(3)) - (1 - sqrt(3))^n*(9 + 2*sqrt(3)) + 3*((1 - sqrt(3))^n + (1 + sqrt(3))^n)*n) for n > 0 (conjectured). - _Colin Barker_, Nov 17 2017

%F a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n > 4 (conjectured). - _Colin Barker_, Nov 17 2017

%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 0,0,0,4,20,80.

%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: A074358 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,1) ) ; end: for n from 0 to 30 do printf("%d,", A074358(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, 1];

%t Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Nov 17 2017 *)

%Y Coefficient of q^0, q^2 and q^3 are in A002605, A074359 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

%K nonn

%O 0,4

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

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