A291726 - OEIS

%I #4 Sep 11 2017 12:06:55

%S 3,6,13,27,51,94,171,303,527,906,1539,2586,4308,7122,11692,19077,

%T 30957,49986,80349,128628,205146,326058,516594,816076,1285674,2020380,

%U 3167464,4954887,7734993,12051616,18743037,29099781,45106223,69810162,107887629,166505313

%N p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = (1 - S)^3.

%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="/A291726/b291726.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 4, -6, 3, -3, 3, 0, 1)

%F G.f.: -(((1 + x^2) (3 - 3 x + x^2 - 3 x^3 + 2 x^4 + x^6))/(-1 + x + x^3)^3).

%F a(n) = 3*a(n-1) - 3*a(n-2) + 4*a(n-3) - 6*a(n-4) + 3*a(n-5) - 3*a(n-6) + 3*a(n-7) + a(n-9) for n >= 10.

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

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

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291726 *)

%Y Cf. A154272, A291728.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Sep 08 2017