%I #31 May 08 2023 09:35:11

%S 0,1,2,6,14,30,62,126,254,510,1022,2046,4094,8190,16382,32766,65534,

%T 131070,262142,524286,1048574,2097150,4194302,8388606,16777214,

%U 33554430,67108862,134217726,268435454,536870910,1073741822,2147483646,4294967294,8589934590,17179869182

%N Expansion of x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).

%C a(n) = A000225(n)-1, a(0)=0, a(1)=1. Number of quasilinear 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 quasilinear weak ordering R.

%C Essentially the same as A095121 and A000918. - _R. J. Mathar_, Jan 02 2018

%H Colin Barker, <a href="/A296965/b296965.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Devillet, <a href="https://arxiv.org/abs/1712.07856">Bisymmetric and quasitrivial operations: characterizations and enumerations</a>, arXiv:1712.07856 [math.RA], 2017-2018.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

%F From _Colin Barker_, Dec 22 2017: (Start)

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

%F a(n) = 2^n - 2 for n>1.

%F a(n) = 3*a(n-1) - 2*a(n-2) for n>3. (End)

%F a(n) = A134067(n-2) for n >= 3. - _Georg Fischer_, Oct 30 2018

%F E.g.f.: 1 + exp(x)*(exp(x) - 2) + x. - _Stefano Spezia_, May 07 2023

%t CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* or *)

%t LinearRecurrence[{3, -2}, {0, 1, 2, 6}, 34] (* _Michael De Vlieger_, Dec 22 2017 *)

%o (PARI) concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ _Colin Barker_, Dec 22 2017

%Y Cf. A000225, A000918, A095121, A134067.

%K nonn,easy

%O 0,3

%A _J. Devillet_, Dec 22 2017