cp's OEIS Frontend

A290901 p-INVERT of the positive integers, where p(S) = 1 - S^3 - S^4.

%I A290901 #10 Aug 18 2017 18:48:30
%S A290901 0,0,1,7,29,93,260,689,1845,5150,14897,43663,127451,368383,1056682,
%T A290901 3022366,8651672,24818978,71319058,205070493,589550733,1694075057,
%U A290901 4866102091,13975547842,40139685023,115298782211,331216158188,951506566087,2733431466995
%N A290901 p-INVERT of the positive integers, where p(S) = 1 - S^3 - S^4.
%C A290901 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 A290901 See A290890 for a guide to related sequences.
%H A290901 Clark Kimberling, <a href="/A290901/b290901.txt">Table of n, a(n) for n = 0..1000</a>
%H A290901 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 57, -71, 57, -28, 8, -1)
%F A290901 a(n) = 8*a(n-1) - 28*a(n-2) + 57*a(n-3) - 71*a(n-4) + 57*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
%F A290901 G.f.: x^2*(1 - x + x^2) / (1 - 8*x + 28*x^2 - 57*x^3 + 71*x^4 - 57*x^5 + 28*x^6 - 8*x^7 + x^8). - _Colin Barker_, Aug 18 2017
%t A290901 z = 60; s = x/(1 - x)^2; p = 1 - s^3 - s^4;
%t A290901 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t A290901 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290901 *)
%o A290901 (PARI) concat(vector(2), Vec(x^2*(1 - x + x^2) / (1 - 8*x + 28*x^2 - 57*x^3 + 71*x^4 - 57*x^5 + 28*x^6 - 8*x^7 + x^8) + O(x^40))) \\ _Colin Barker_, Aug 18 2017
%Y A290901 Cf. A000027, A290890.
%K A290901 nonn,easy
%O A290901 0,4
%A A290901 _Clark Kimberling_, Aug 17 2017