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p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4 - S^5.
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%I #11 Jun 02 2023 21:51:09

%S 1,3,9,27,81,242,720,2137,6337,18789,55715,165232,490058,1453493,

%T 4311025,12786359,37923789,112480082,333610072,989469949,2934716101,

%U 8704215281,25816251319,76569665176,227101665034,673571786617,1997779058053,5925309279179

%N p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4 - S^5.

%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 A291000 for a guide to related sequences.

%H Clark Kimberling, <a href="/A291007/b291007.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,14,-7,2).

%F a(n) = 6*a(n-1) - 13*a(n-2) + 14*a(n-3) - 7*a(n-4) + 2*a(n-5) for n >= 6.

%F G.f.: (1 - 3*x + 4*x^2 - 2*x^3 + x^4) / (1 - 6*x + 13*x^2 - 14*x^3 + 7*x^4 - 2*x^5). - _Colin Barker_, Aug 23 2017

%t z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3 - s^4 - s^5;

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291007 *)

%t LinearRecurrence[{6,-13,14,-7,2},{1,3,9,27,81},30] (* _Harvey P. Dale_, Apr 07 2019 *)

%o (PARI) Vec((1 -3*x +4*x^2 -2*x^3 +x^4)/(1 -6*x +13*x^2 -14*x^3 +7*x^4 - 2*x^5) + O(x^30)) \\ _Colin Barker_, Aug 23 2017

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1 -3*x +4*x^2 -2*x^3 +x^4)/(1 -6*x +13*x^2 -14*x^3 +7*x^4 -2*x^5) )); // _G. C. Greubel_, Jun 01 2023

%o (SageMath)

%o def A291007_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1 -3*x +4*x^2 -2*x^3 +x^4)/(1 -6*x +13*x^2 -14*x^3 +7*x^4 -2*x^5) ).list()

%o A291007_list(40) # _G. C. Greubel_, Jun 01 2023

%Y Cf. A000012, A289780, A291000.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Aug 23 2017