A290926 - OEIS

%I #6 Aug 19 2017 13:26:30

%S 0,2,8,23,64,182,520,1475,4152,11624,32408,90028,249272,688140,

%T 1894600,5203665,14260968,39004962,106486512,290226621,789776888,

%U 2146082610,5823823120,15784464728,42731452816,115556460982,312175750152,842537682283,2271900155120

%N p-INVERT of the positive integers, where p(S) = (1 - S^2)^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 A290890 for a guide to related sequences.

%H Clark Kimberling, <a href="/A290926/b290926.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 (8, -26, 48, -59, 48, -26, 8, -1)

%F G.f.: (2 x - 8 x^2 + 11 x^3 - 8 x^4 + 2 x^5)/(1 - 4 x + 5 x^2 - 4 x^3 + x^4)^2.

%F a(n) = 8*a(n-1) - 26*a(n-2) + 48*a(n-3) - 59*a(n-4) + 48*a(n-5) - 26*a(n-6) + 8*a(n-7) - a(n-8).

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

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)

%t u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290926 *)

%Y Cf. A000027, A290890.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Aug 19 2017