A291741 - OEIS

%I #4 Sep 14 2017 18:15:13

%S 1,0,1,1,1,4,5,7,11,12,19,30,42,68,98,137,205,292,429,644,936,1380,

%T 2024,2936,4316,6324,9260,13625,19949,29216,42841,62701,91917,134784,

%U 197485,289547,424331,621708,911255,1335378,1957086,2868620,4203998,6161329

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

%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="/A291741/b291741.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 (1, -1, 2, -2, 3, -1, 3, 0, 1)

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

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

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

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

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291741 *)

%Y Cf. A154272, A291728.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Sep 12 2017