A074086 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)=(2,1).

0, 0, 0, 0, 0, 14, 71, 282, 997, 3298, 10439, 32012, 95834, 281494, 814131, 2324422, 6564135, 18362810, 50947395, 140329400, 384031508, 1044880222, 2828084399, 7618214354, 20432838121, 54585196818, 145287466799, 385397215108

Offset: 0

Author

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

Keywords

• nonn

Comments

The coefficient of q^0 is the Pell number A000129(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^3 are 0,0,0,0,0,14.
		

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 (8, -20, 8, 26, -8, -20, -8, -1).

Crossrefs

Coefficients of q^0, q^1 and q^2 are in A000129, A074084 and A074085. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

Programs

Mathematica

b=2; 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[{8, -20, 8, 26, -8, -20, -8, -1}, {0, 0, 14, 71, 282, 997, 3298, 10439}, 25]] (* Jean-François Alcover, Jan 27 2019 *)

Formula

G.f.: (14x^5-41x^6-6x^7+49x^8+30x^9+5x^10)/(1-2x-x^2)^4.

a(n) = 8a(n-1)-20a(n-2)+8a(n-3)+26a(n-4)-8a(n-5)-20a(n-6)-8a(n-7)-a(n-8) for n>=11.

Extensions

Edited by Dean Hickerson, Aug 21 2002