cp's OEIS Frontend

A291016 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 4*S + S^2.

%I A291016 #11 Mar 14 2025 02:34:58
%S A291016 4,19,90,426,2016,9540,45144,213624,1010880,4783536,22635936,
%T A291016 107114400,506870784,2398538304,11350005120,53708800896,254152774656,
%U A291016 1202663842560,5691066407424,26930415389184,127436093890560,603034071008256,2853587862706176
%N A291016 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 4*S + S^2.
%C A291016 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 A291016 See A291000 for a guide to related sequences.
%H A291016 Clark Kimberling, <a href="/A291016/b291016.txt">Table of n, a(n) for n = 0..1000</a>
%H A291016 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).
%F A291016 G.f.: (4 - 5*x)/(1 - 6*x + 6*x^2).
%F A291016 a(n) = 6*a(n-1) - 6*a(n-2) n >= 3.
%F A291016 From _G. C. Greubel_, Jun 06 2023: (Start)
%F A291016 a(n) = ((3+sqrt(3))^(n+4) - (3-sqrt(3))^(n+4))/(72*sqrt(3)).
%F A291016 a(n) = 6^(n/2)*(4*ChebyshevU(n, sqrt(3/2)) - (5/sqrt(6))*ChebyshevU(n - 1, sqrt(3/2))).
%F A291016 E.g.f.: exp(3*x)*(4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x)). (End)
%t A291016 z = 60; s = x/(1-x); p = 1 - 4 s + s^2;
%t A291016 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291016 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291016 *)
%t A291016 LinearRecurrence[{6,-6}, {4,19}, 40] (* _G. C. Greubel_, Jun 06 2023 *)
%o A291016 (Magma) I:=[4,19]; [n le 2 select I[n] else 6*(Self(n-1)-Self(n-2)): n in [1..40]]; // _G. C. Greubel_, Jun 06 2023
%o A291016 (SageMath)
%o A291016 A291016=BinaryRecurrenceSequence(6,-6,4,19)
%o A291016 [A291016(n) for n in range(51)] # _G. C. Greubel_, Jun 06 2023
%Y A291016 Cf. A000012, A033453, A289780, A291000.
%K A291016 nonn,easy
%O A291016 0,1
%A A291016 _Clark Kimberling_, Aug 23 2017