A293004 Expansion of 2*x^2 / (x^3 + x^2 - 3x + 1).

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

Offset: 0

Author

J. Devillet, Sep 28 2017

Keywords

• nonn
• easy

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Andrei Asinowski, Cyril Banderier, Sara Billey, Benjamin Hackl, Svante Linusson, Pop-stack sorting and its image: permutations with overlapping runs, Eurocomb, Acta Math. Univ. Comenianae (2019), 1-8.
• M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Crossrefs

Cf. A000129, A048739.

Essentially the same as A265278.

Programs

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

Maple

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

Mathematica

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 *)

PARI

concat(vector(2), Vec(2*x^2 / (x^3+x^2-3*x+1) + O(x^40))) \\ Colin Barker, Sep 28 2017

Formula

G.f.: 2*x^2 / (x^3 + x^2 - 3x + 1).

a(n) = 2*A048739(n-2), a(0) = a(1) = 0.

From Colin Barker, Sep 28 2017: (Start)

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)

a(n) = A265278(n) for n != 1. - Joerg Arndt, Oct 01 2017