login
A296965
Expansion of x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).
2
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
0,3
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<...<n and for which {1,...,n} has exactly one maximal element for the quasilinear weak ordering R.
Essentially the same as A095121 and A000918. - R. J. Mathar, Jan 02 2018
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
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 *)
PROG
(PARI) concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. Devillet, Dec 22 2017
STATUS
approved