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A016137
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Expansion of 1/((1-3x)(1-6x)).
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5
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1, 9, 63, 405, 2511, 15309, 92583, 557685, 3352671, 20135709, 120873303, 725416965, 4353033231, 26119793709, 156723545223, 940355620245, 5642176768191, 33853189749309, 203119525916343, 1218718317759525
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = (3^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = -3^n + 2*6^n.
E.g.f.: (d^2/dx^2)((((exp(3*x)-1)/3)^2)/2!) = -exp(3*x) + 2*exp(6*x).
With leading zero, this is (6^n - 3^n)/3, the binomial transform of A016127 (with extra leading zero). - Paul Barry, Aug 20 2003
With leading zero, this is the fourth binomial transform of A001045, with a(n) = (2^n-1)(3^n/3 - 0^n/3) = A000225(n)*(A000244(n-1) - 0^n/3). - Paul Barry, Apr 28 2004
Sum_{k=1..n} 3^(k-1)*3^(n-k)*binomial(n, k). - Zerinvary Lajos, Sep 24 2006
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MATHEMATICA
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CoefficientList[Series[1/((1-3x)(1-6x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{9, -18}, {1, 9}, 30] (* Harvey P. Dale, Jul 07 2012 *)
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PROG
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(Sage) [lucas_number1(n, 9, 18) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009
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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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STATUS
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approved
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