A296965 Expansion of x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).

0, 1, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182

Offset: 0

Author

J. Devillet, Dec 22 2017

Keywords

• nonn
• easy

Comments

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<...

Essentially the same as A095121 and A000918. - R. J. Mathar, Jan 02 2018

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017-2018.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A000225, A000918, A095121, A134067.

Programs

Mathematica

CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* or *)
LinearRecurrence[{3, -2}, {0, 1, 2, 6}, 34] (* Michael De Vlieger, Dec 22 2017 *)

PARI

concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

Formula

From Colin Barker, Dec 22 2017: (Start)

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

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

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

a(n) = A134067(n-2) for n >= 3. - Georg Fischer, Oct 30 2018

E.g.f.: 1 + exp(x)*(exp(x) - 2) + x. - Stefano Spezia, May 07 2023

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