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A074355 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)=(1,3).

%I A074355 #19 Nov 23 2017 15:47:50
%S A074355 0,0,0,3,15,45,147,402,1134,2991,7917,20367,52167,131748,330876,
%T A074355 824187,2042763,5035473,12361755,30226614,73664298,178971879,
%U A074355 433649769,1048133619,2527706127,6083434824,14613750648,35045236083,83909261319
%N A074355 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)=(1,3).
%C A074355 Coefficient of q^0 is A006130.
%H A074355 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 A074355 G.f.: (9x^4+3x^3)/(1-3x-3x^2)^2 (conjectured). - _Ralf Stephan_, May 09 2004
%e A074355 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^1 are 0,0,0,3,15,45.
%p A074355 nu := proc(n,b,lambda) option remember ; 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:
%p A074355 A074355 := proc(n) local b,lambda,thisnu ; b := 1 ; lambda := 3 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,1) ) ; end: # _R. J. Mathar_, Mar 20 2007
%t A074355 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 A074355 a[n_] := a[n] = Coefficient[nu[n, 1, 3], q, 1];
%t A074355 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 30}] (* _Jean-François Alcover_, Nov 23 2017, from 1st Maple program *)
%Y A074355 Coefficient of q^0, q^2 and q^3 are in A006130, A074356 and A074357. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074354, A074358-A074363.
%K A074355 nonn
%O A074355 0,4
%A A074355 Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%E A074355 More terms from _R. J. Mathar_, Mar 20 2007