A074089 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)=(2,3).

0, 0, 0, 0, 0, 78, 501, 2574, 11757, 50034, 203229, 797316, 3046362, 11394774, 41885913, 151732722, 542840175, 1921208586, 6735519249, 23417342568, 80810560596, 277008392478, 943826398893, 3198199361910, 10783017814065

Offset

0,6

Comments

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

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 (8,-12,-40,74,120,-108,-216,-81).

Formula

G.f.: (78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4.

a(n) = 8*a(n-1) -12*a(n-2) -40*a(n-3) +74*a(n-4) +120*a(n-5) -108*a(n-6) -216*a(n-7) -81*a(n-8) for n>=11.

Example

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^3 are 0,0,0,0,0,78.

Mathematica

b=2; lambda=3; 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: *)
CoefficientList[Series[(78*x^5-123*x^6-498*x^7+297*x^8+1134*x^9 + 567*x^10)/( 1-2*x-3*x^2)^4, {x, 0, 50}], x] (* G. C. Greubel, May 26 2018 *)

Prog

(PARI) x='x+O('x^30); concat([0, 0, 0, 0, 0], Vec((78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4)) \\ G. C. Greubel, May 26 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0] cat Coefficients(R!((78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4)); // G. C. Greubel, May 26 2018

Crossrefs

Coefficients of q^0, q^1 and q^2 are in A014983, A074087 and A074088. Related sequences with other values of b and lambda are in A074082-A074086.

Sequence in context: A007255 A003913 A251321 * A117329 A057798 A057800

Adjacent sequences: A074086 A074087 A074088 * A074090 A074091 A074092

Keyword

nonn,easy

Author

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

Extensions

Edited by Dean Hickerson, Aug 21 2002

Status

approved