%I #8 Aug 25 2017 06:23:09

%S 4,10,24,55,120,254,524,1059,2104,4120,7968,15244,28888,54284,101240,

%T 187537,345268,632122,1151408,2087485,3768280,6775322,12136940,

%U 21666712,38555100,68401582,121011800,213521067,375813760,659910710,1156204452,2021495767

%N p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S)^4.

%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 A291219 for a guide to related sequences.

%H Clark Kimberling, <a href="/A291224/b291224.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -8, 5, 8, -2, -4, -1)

%F a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) for n >= 9.

%F G.f.: (2 - x - 2*x^2)*(2 - 2*x - 3*x^2 + 2*x^3 + 2*x^4) / (1 - x - x^2)^4. - _Colin Barker_, Aug 25 2017

%t z = 60; s = x/(1 - x^2); p = (1 - s)^4;

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291224 *)

%o (PARI) Vec((2 - x - 2*x^2)*(2 - 2*x - 3*x^2 + 2*x^3 + 2*x^4) / (1 - x - x^2)^4 + O(x^40)) \\ _Colin Barker_, Aug 25 2017

%Y Cf. A000035, A291219.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Aug 24 2017