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p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.
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%I #10 Apr 15 2023 02:04:14

%S 0,0,1,3,6,12,27,63,143,315,684,1479,3195,6903,14932,32361,70266,

%T 152775,332397,723330,1573829,3423444,7444722,16185939,35185779,

%U 76483890,166253545,361396431,785621808,1707884880,3712912632,8071922817,17548551692,38150905170

%N p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.

%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="/A290997/b290997.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,21,-18,9,-1).

%F a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 18*a(n-4) + 9*a(n-5) - a(n-6) for n >= 7.

%F G.f.: x^2*(1 - 3*x + 3*x^2) / (1 - 6*x + 15*x^2 - 21*x^3 + 18*x^4 - 9*x^5 + x^6). - _Colin Barker_, Aug 22 2017

%t z = 60; s = x/(1-x); p= 1 -s^3 -s^6;

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

%t Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1] (* A290997 *)

%t LinearRecurrence[{6,-15,21,-18,9,-1}, {0,0,1,3,6,12}, 40] (* _G. C. Greubel_, Apr 14 2023 *)

%o (PARI) concat(vector(2), Vec(x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) + O(x^50))) \\ _Colin Barker_, Aug 22 2017

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) )); // _G. C. Greubel_, Apr 14 2023

%o (SageMath)

%o def A290997_list(prec):

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

%o return P( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) ).list()

%o A290997_list(40) # _G. C. Greubel_, Apr 14 2023

%Y Cf. A000012, A033453, A289780, A291000.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Aug 22 2017