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%I #9 Aug 15 2017 19:55:03
%S 1,6,29,133,597,2661,11856,52878,235986,1053345,4701627,20985035,
%T 93662073,418038721,1865820223,8327671681,37168717729,165894342774,
%U 740432630793,3304756826019,14750048986898,65833571645931,293833543748968,1311460845206801
%N p-INVERT of the tetrahedral numbers (A000292), where p(S) = 1 - S - 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 A289780 for a guide to related sequences.
%H Clark Kimberling, <a href="/A289801/b289801.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 (9, -31, 62, -74, 57, -28, 8, -1)
%F G.f.: (1 - 3 x + 6 x^2 - 4 x^3 + x^4)/(1 - 9 x + 31 x^2 - 62 x^3 + 74 x^4 - 57 x^5 + 28 x^6 - 8 x^7 + x^8).
%F a(n) = 9*a(n-1) - 31*a(n-2) + 62*a(n-3) - 74*a(n-4) + 57*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
%t z = 60; s = x/(1 - x)^4; p = 1 - s - s^2;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000292 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289801 *)
%Y Cf. A000292, A289780.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Aug 12 2017
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