A291142 - OEIS

%I #9 Feb 10 2018 19:42:37

%S 0,1,2,6,16,43,114,300,784,2037,5266,13554,34752,88799,226210,574680,

%T 1456352,3682409,9292002,23403102,58842416,147713043,370262930,

%U 926852868,2317191024,5786293597,14433117938,35964267594,89528469088,222666575815,553319176770

%N a(n) = (1/4)*A291024(n).

%C Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

%C See A291000 for a guide to related sequences.

%H Clark Kimberling, <a href="/A291142/b291142.txt">Table of n, a(n) for n = 0..999</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -4, -1)

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

%F a(n) = 4*a(n-1) - 2 a(n-2) - 4*a(n-3) - a(n-4) for n >= 5.

%F a(n) = (1/4)*A291024(n) for n >= 0.

%F a(n) = ((1+sqrt(2))^n*(3*sqrt(2) + 2*(-1+sqrt(2))*n) - (1-sqrt(2))^n*(3*sqrt(2) + 2*(1+sqrt(2))*n)) / 16. - _Colin Barker_, Aug 24 2017

%t z = 60; s = x/(1 - x); p = 1 - 3 s^2 + 2 s^3;

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291024 *)

%t u/4 (* A291142 *)

%o (PARI) concat(0, Vec(x*(1 - 2*x) / (1 - 2*x - x^2)^2 + O(x^40))) \\ _Colin Barker_, Aug 24 2017

%Y Cf. A000012, A289780, A291000.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Aug 24 2017