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A292494 p-INVERT of the odd positive integers, where p(S) = 1 - S - S^2 - S^3.

%I A292494 #4 Oct 05 2017 09:50:57
%S A292494 1,5,21,88,362,1470,5940,23996,97028,392592,1588840,6430088,26021472,
%T A292494 105301184,426118816,1724362608,6977946160,28237566352,114268643984,
%U A292494 462409605552,1871227376592,7572272759344,30642622403664,124001121308400,501793808163600
%N A292494 p-INVERT of the odd positive integers, where p(S) = 1 - S - S^2 - S^3.
%C A292494 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 A292494 See A292480 for a guide to related sequences.
%H A292494 Clark Kimberling, <a href="/A292494/b292494.txt">Table of n, a(n) for n = 0..1000</a>
%H A292494 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7, -17, 23, -12, 6, 2)
%F A292494 G.f.: -(((1 + x) (1 - 3 x + 6 x^2 - 3 x^3 + 3 x^4))/(-1 + 7 x - 17 x^2 + 23 x^3 - 12 x^4 + 6 x^5 + 2 x^6)).
%F A292494 a(n) = 7*a(n-1) - 17*a(n-2) + 23*a(n-3) + 12*a(n-4) + 6*a(n-5) + 2*a(n-6) for n >= 7.
%t A292494 z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s - s^2 - s^3;
%t A292494 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
%t A292494 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292494 *)
%Y A292494 Cf. A005408, A292480.
%K A292494 nonn,easy
%O A292494 0,2
%A A292494 _Clark Kimberling_, Oct 04 2017