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Inverse binomial transform of A140359.
4

%I #23 Jun 29 2023 23:28:23

%S 1,0,5,-5,15,-25,55,-105,215,-425,855,-1705,3415,-6825,13655,-27305,

%T 54615,-109225,218455,-436905,873815,-1747625,3495255,-6990505,

%U 13981015,-27962025,55924055,-111848105,223696215,-447392425,894784855,-1789569705,3579139415

%N Inverse binomial transform of A140359.

%C For p*Jacobsthal numbers A001045, p=2: A078008 (A001045 differences, they are companions) or 1, 2*A001045(n), also in A133494; p=3: A062510; p=4: see A097073; p=6: A092297.

%H Alois P. Heinz, <a href="/A140360/b140360.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1, 2).

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

%F a(n) = (-5*(-1 + (-2)^(n-1)))/3, for n>0. - _Andres Cicuttin_, Apr 15 2016

%F a(n) = 5 - 2*a(n-1), for n>2. - _Andres Cicuttin_, Apr 15 2016

%p a:= n-> `if`(n=0, 1, (<<0|1>, <2|-1>>^(n-1). <<0,5>>)[1,1]):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Dec 28 2010

%t {1}~Join~Table[(-5 (-1 + (-2)^(n - 1)))/3, {n, 32}] (* or *)

%t CoefficientList[Series[(-3 x^2 - x - 1)/(2 x^2 - x - 1), {x, 0, 32}], x] (* _Michael De Vlieger_, Apr 15 2016 *)

%K sign

%O 0,3

%A _Paul Curtz_, Jun 24 2008

%E More terms from _Alois P. Heinz_, Dec 28 2010