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A291031 p-INVERT of the positive integers, where p(S) = 1 - 3*S + 2*S^3. 2

%I

%S 3,15,70,321,1461,6624,29967,135399,611318,2758881,12447753,56154744,

%T 253306119,1142572767,5153589754,23244956169,104843981505,

%U 472885383744,2132882300571,9620044596687,43389716584682,195702453488433,882684641446989,3981207177094608

%N p-INVERT of the positive integers, where p(S) = 1 - 3*S + 2*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 A290890 for a guide to related sequences.

%H Clark Kimberling, <a href="/A291031/b291031.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 (9, -27, 36, -27, 9, -1)

%F G.f.: (3 - 12 x + 16 x^2 - 12 x^3 + 3 x^4)/(1 - 9 x + 27 x^2 - 36 x^3 + 27 x^4 - 9 x^5 + x^6).

%F a(n) = 9*a(n-1) - 27*a(n-2) + 36*a(n-3) - 27*a(n-4) + 90*a(n-5) - a(n-6).

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

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

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291031 *)

%Y Cf. A000027, A290890.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Aug 19 2017

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Last modified February 19 07:51 EST 2019. Contains 320309 sequences. (Running on oeis4.)