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A291014 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^3)^2.

%I A291014 #10 Jun 06 2023 08:18:05
%S A291014 0,0,2,6,12,23,48,105,228,486,1026,2161,4548,9555,20026,41874,87384,
%T A291014 182043,378648,786429,1631120,3378750,6990510,14447045,29826156,
%U A291014 61516455,126761190,260978922,536870916,1103567983,2266788288,4652881233,9544371772,19565962134
%N A291014 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^3)^2.
%C A291014 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 A291014 See A291000 for a guide to related sequences.
%H A291014 Clark Kimberling, <a href="/A291014/b291014.txt">Table of n, a(n) for n = 0..1000</a>
%H A291014 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,22,-21,12,-4).
%F A291014 G.f.: x^2*(2 - 6*x + 6*x^2 - 3*x^3)/( (1-2*x)*(1-x+x^2) )^2.
%F A291014 a(n) = 6*a(n-1) - 15*a(n-2) + 22*a(n-3) - 21*a(n-4) + 12*a(n-5) - 4*a(n-6) for n >= 7.
%F A291014 a(n) = (1/9)*( 2^(n-1)*(n + 8) - 3*(A099254(n) - A099254(n-1)) - A010892(n) - 5*A010892(n-1) ). - _G. C. Greubel_, Jun 05 2023
%t A291014 z = 60; s = x/(1-x); p = (1 - s^3)^2;
%t A291014 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291014 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291014 *)
%t A291014 LinearRecurrence[{6,-15,22,-21,12,-4}, {0,0,2,6,12,23}, 50] (* _G. C. Greubel_, Jun 05 2023 *)
%o A291014 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0,0] cat Coefficients(R!( x^2*(2-6*x+6*x^2-3*x^3)/((1-2*x)*(1-x+x^2))^2 )); // _G. C. Greubel_, Jun 05 2023
%o A291014 (SageMath)
%o A291014 def A291014_list(prec):
%o A291014     P.<x> = PowerSeriesRing(ZZ, prec)
%o A291014     return P( x^2*(2-6*x+6*x^2-3*x^3)/((1-2*x)*(1-x+x^2))^2 ).list()
%o A291014 A291014_list(50) # _G. C. Greubel_, Jun 05 2023
%Y A291014 Cf. A000012, A010892, A033453, A099254, A289780, A291000.
%K A291014 nonn,easy
%O A291014 0,3
%A A291014 _Clark Kimberling_, Aug 23 2017