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A293004
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Expansion of 2*x^2 / (x^3 + x^2 - 3x + 1).
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4
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0, 0, 2, 6, 16, 40, 98, 238, 576, 1392, 3362, 8118, 19600, 47320, 114242, 275806, 665856, 1607520, 3880898, 9369318, 22619536, 54608392, 131836322, 318281038, 768398400, 1855077840, 4478554082, 10812186006, 26102926096, 63018038200, 152139002498
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OFFSET
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0,3
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COMMENTS
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Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one maximal element for the weak ordering R.
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LINKS
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FORMULA
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G.f.: 2*x^2 / (x^3 + x^2 - 3x + 1).
a(n) = 2*A048739(n-2), a(0) = a(1) = 0.
a(n) = 3*a(n-1) - a(n-2) - a(n-3) for n > 2.
a(n) = (-2 + (1-sqrt(2))^n + (1+sqrt(2))^n) / 2. (End)
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MAPLE
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A293004:=gfun:-rectoproc({a(n)=3*a(n-1) -a(n-2)-a(n-3), a(0)=0, a(1)=0, a(2)=2}, a(n), remember): map(A293004, [$0..10^3]); # Muniru A Asiru, Oct 09 2017
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MATHEMATICA
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CoefficientList[Series[2 x^2/(x^3 + x^2 - 3 x + 1), {x, 0, 30}], x] (* Michael De Vlieger, Oct 06 2017 *)
RecurrenceTable[{a[1]==a[2]==0, a[3]==2, a[n]==3a[n-1] - a[n-2] - a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Oct 09 2017 *)
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PROG
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(PARI) concat(vector(2), Vec(2*x^2 / (x^3+x^2-3*x+1) + O(x^40))) \\ Colin Barker, Sep 28 2017
(Magma) I:=[0, 0, 2]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Oct 09 2017
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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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