login
p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = (1 - S)(1 - 2 S).
2

%I #4 Sep 11 2017 20:05:03

%S 3,7,18,45,108,258,615,1459,3453,8164,19287,45540,107496,253695,

%T 598659,1412587,3332970,7863853,18553752,43774722,103279023,243668295,

%U 574890057,1356344056,3200033343,7549859464,17812425600,42024945087,99149648967,233924207559

%N p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = (1 - S)(1 - 2 S).

%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 A291728 for a guide to related sequences.

%H Clark Kimberling, <a href="/A291734/b291734.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2, 3, -4, 0, -2)

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

%F a(n) = 3*a(n-1) - 2*a(n-2) + 3*a(n-3) - 4*a(n-4) - 2*a(n-6) for n >= 7.

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

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A154272 *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291734 *)

%Y Cf. A154272, A291728.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Sep 11 2017