%I #19 Nov 23 2017 15:51:39
%S 0,0,0,0,12,42,180,561,1833,5373,15798,44367,123561,336243,906054,
%T 2408094,6344832,16561824,42922602,110472933,282678423,719404803,
%U 1822117962,4594816221,11540742615,28880919975,72033463644,179107709004
%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)=(1,3).
%C Coefficient of q^0 is A006130.
%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 Conjectures from _Colin Barker_, Nov 18 2017: (Start)
%F G.f.: 3*x^4*(2 - 3*x)*(2 + 4*x + 3*x^2) / (1 - x - 3*x^2)^3.
%F a(n) = 3*a(n-1) + 6*a(n-2) - 17*a(n-3) - 18*a(n-4) + 27*a(n-5) + 27*a(n-6) for n>7.
%F (End)
%e The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=4, nu(3)=7+3q, nu(4)=19+15q+12q^2, nu(5)=40+45q+42q^2+30q^3+9q^4, so the coefficients of q^2 are 0,0,0,0,12,42.
%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: A074356 := proc(n) local b,lambda,thisnu ; b := 1 ; lambda := 3 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074356(n) ) ; od ; # _R. J. Mathar_, Mar 20 2007
%t nu[n_, b_, lambda_] := nu[n, b, lambda] = Which[n == 0, 1, n == 1, b, True, b*nu[n - 1, b, lambda] + lambda*nu[n - 2, b, lambda]*Sum[q^i, {i, 0, n - 2}]];
%t a[n_] := a[n] = Coefficient[nu[n, 1, 3], q, 2];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 30}] (* _Jean-François Alcover_, Nov 23 2017, from Maple *)
%Y Coefficient of q^0, q^1 and q^3 are in A006130, A074355 and A074357. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074354, A074358-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