A074088 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)=(2,3).

0, 0, 0, 0, 21, 120, 585, 2508, 10122, 39042, 145974, 532704, 1907451, 6725004, 23407287, 80591148, 274899288, 930128646, 3124838844, 10432356000, 34634029713, 114403303008, 376184538165, 1231890463020, 4018920819606

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The coefficient of q^0 is A014983(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=7, nu(3)=20+6q, nu(4)=61+33q+21q^2, nu(5)=182+144q+120q^2+78q^3+18q^4, so the coefficients of q^2 are 0,0,0,0,21,120.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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 (6,-3,-28,9,54,27).

Crossrefs

Coefficients of q^0, q^1 and q^3 are in A014983, A074087 and A074089. Related sequences with other values of b and lambda are in A074082-A074086.

Programs

Magma

I:=[0,0,21,120,585,2508]; [0,0] cat [n le 6 select I[n] else 6*Self(n-1) -3*Self(n-2) -28*Self(n-3) +9*Self(n-4) +54*Self(n-5) +27*Self(n-6): n in [1..30]]; // G. C. Greubel, May 26 2018

Mathematica

b=2; lambda=3; expon=2; 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},LinearRecurrence[{6,-3,-28,9,54,27},{0,0,21,120,585,2508},40]] (* Harvey P. Dale, Apr 28 2012 *)

PARI

x='x+O('x^30); concat([0,0,0,0], Vec((21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3)) \\ G. C. Greubel, May 26 2018

Formula

G.f.: (21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3.

a(n) = 6*a(n-1) -3*a(n-2) -28*a(n-3) +9*a(n-4) +54*a(n-5) +27*a(n-6) for n>=8.

Extensions

Edited by Dean Hickerson, Aug 21 2002