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A074088
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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).
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3
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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
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OFFSET
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0,5
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COMMENTS
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The coefficient of q^0 is A014983(n+1).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002
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EXTENSIONS
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STATUS
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approved
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