A074084 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)=(2,1).

0, 0, 0, 2, 9, 32, 102, 306, 883, 2480, 6828, 18514, 49597, 131568, 346194, 904738, 2350695, 6076960, 15641304, 40103778, 102473969, 261046144, 663180222, 1680628946, 4249496795, 10722962256, 27007159428, 67904097074

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..27.

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, -4, -1).

Formula

G.f.: (2x^3+x^4)/(1-2x-x^2)^2.

a(n) = 4a(n-1)-2a(n-2)-4a(n-3)-a(n-4) for n>=5.

Example

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^1 are 0,0,0,2,9,32.

Mathematica

b=2; lambda=1; expon=1; 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}, LinearRecurrence[{4, -2, -4, -1}, {0, 0, 2, 9}, 30]] (* Harvey P. Dale, Apr 18 2012 *)

Crossrefs

Coefficients of q^0, q^2 and q^3 are in A000129, A074085 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

Sequence in context: A198016 A082114 A332870 * A053152 A077644 A292482

Adjacent sequences: A074081 A074082 A074083 * A074085 A074086 A074087

Keyword

nonn

Author

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

Extensions

Edited by Dean Hickerson, Aug 21 2002

Status

approved