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%I #31 Sep 08 2022 08:45:06
%S 0,0,0,0,21,120,585,2508,10122,39042,145974,532704,1907451,6725004,
%T 23407287,80591148,274899288,930128646,3124838844,10432356000,
%U 34634029713,114403303008,376184538165,1231890463020,4018920819606
%N 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).
%C The coefficient of q^0 is A014983(n+1).
%H G. C. Greubel, <a href="/A074088/b074088.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Beattie, S. Dăscălescu and S. Raianu, <a href="https://arxiv.org/abs/math/0204075">Lifting of Nichols Algebras of Type B_2</a>, arXiv:math/0204075 [math.QA], 2002.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-3,-28,9,54,27).
%F G.f.: (21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3.
%F 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.
%e 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.
%t 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]
%t (* Second program: *)
%t Join[{0,0},LinearRecurrence[{6,-3,-28,9,54,27},{0,0,21,120,585,2508},40]] (* _Harvey P. Dale_, Apr 28 2012 *)
%o (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
%o (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
%Y 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.
%K nonn,easy
%O 0,5
%A Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002
%E Edited by _Dean Hickerson_, Aug 21 2002
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