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%I #16 Sep 08 2022 08:46:19
%S 1,4,15,56,208,767,2812,10278,37530,137044,500571,1828818,6682264,
%T 24416877,89218462,325997507,1191160160,4352355633,15902968338,
%U 58107491971,212317732888,775783501558,2834620130881,10357363200392,37844566834330,138279520124262
%N p-INVERT of the positive integers, where p(S) = 1 - S - S^2 - S^3 - S^4.
%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="/A291030/b291030.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,-33,68,-85,68,-33,9,-1).
%F G.f.: (1 - 5 x + 12 x^2 - 15 x^3 + 12 x^4 - 5 x^5 + x^6)/(1 - 9 x + 33 x^2 - 68 x^3 + 85 x^4 - 68 x^5 + 33 x^6 - 9 x^7 + x^8).
%F a(n) = 9*a(n-1) - 33*a(n-2) + 68*a(n-3) - 85*a(n-4) + 68*a(n-5) - 33*a(n-6) + 9*a(n-7) - a(n-8).
%t z = 60; s = x/(1 - x)^2; p = 1 - s - s^2 - s^3 - s^4;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291030 *)
%t LinearRecurrence[{9, -33, 68, -85, 68, -33, 9, -1}, {1, 4, 15, 56, 208, 767, 2812, 10278}, 40] (* _Vincenzo Librandi_, Aug 20 2017 *)
%o (Magma) I:=[1,4,15,56,208,767,2812,10278]; [n le 8 select I[n] else 9*Self(n-1)-33*Self(n-2)+68*Self(n-3)-85*Self(n-4)+68*Self(n-5)-33*Self(n-6)+9*Self(n-7)-Self(n-8): n in [1..40]]; // _Vincenzo Librandi_, Aug 20 2017
%Y Cf. A000027, A290890.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Aug 19 2017
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