A074083 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)=(1,1).

0, 0, 0, 0, 0, 4, 14, 39, 97, 224, 494, 1051, 2177, 4412, 8784, 17228, 33360, 63886, 121164, 227833, 425147, 787916, 1451198, 2657821, 4842727, 8782230, 15857426, 28517864, 51095760, 91232520, 162372682, 288115147, 509790277, 899630376

Offset: 0

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

Keywords

• nonn

Comments

The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^3 are 0,0,0,0,0,4.
		

Links

• M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
• Index entries for linear recurrences with constant coefficients, signature (4, -2, -8, 5, 8, -2, -4, -1).

Crossrefs

Coefficients of q^0, q^1 and q^2 are in A000045, A023610 and A074082. Related sequences with different values of b and lambda are in A074084-A074089.

Programs

Mathematica

b=1; lambda=1; expon=3; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0, 0}, LinearRecurrence[{4, -2, -8, 5, 8, -2, -4, -1}, {0, 0, 4, 14, 39, 97, 224, 494}, 31]] (* Jean-François Alcover, Jan 27 2019 *)

Formula

G.f.: (4x^5-2x^6-9x^7+x^8+6x^9+2x^10)/(1-x-x^2)^4.

a(n) = 4a(n-1)-2a(n-2)-8a(n-3)+5a(n-4)+8a(n-5)-2a(n-6)-4a(n-7)-a(n-8) for n>=11.

Extensions

Edited by Dean Hickerson, Aug 21 2002