cp's OEIS Frontend

A291001 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 8*S^2.

%I A291001 #16 Jul 15 2024 06:04:55
%S A291001 0,8,16,88,288,1192,4400,17144,65088,250184,955984,3663256,14018400,
%T A291001 53679592,205487984,786733112,3011882112,11530896008,44144966800,
%U A291001 169006205656,647027178912,2477097797416,9483385847216,36306456276344,138996613483200,532138420900808
%N A291001 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 8*S^2.
%C A291001 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 A291001 See A291000 for a guide to related sequences.
%H A291001 Clark Kimberling, <a href="/A291001/b291001.txt">Table of n, a(n) for n = 0..1000</a>
%H A291001 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,7).
%F A291001 G.f.: 8*x/(1 - 2*x - 7*x^2).
%F A291001 a(n) = 2*a(n-1) + 7*a(n-2) for n >= 3.
%F A291001 a(n) = 8*A015519(n).
%F A291001 a(n) = sqrt(2)*((1+2*sqrt(2))^n - (1-2*sqrt(2))^n). - _Colin Barker_, Aug 23 2017
%t A291001 z = 60; s = x/(1 - x); p = 1 - s^8;
%t A291001 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291001 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291001 *)
%t A291001 LinearRecurrence[{2,7}, {0,8}, 41] (* _G. C. Greubel_, Apr 25 2023 *)
%o A291001 (Magma) [n le 2 select 8*(n-1) else 2*Self(n-1) +7*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Apr 25 2023
%o A291001 (SageMath)
%o A291001 A291001=BinaryRecurrenceSequence(2,7,0,8)
%o A291001 [A291001(n) for n in range(41)] # _G. C. Greubel_, Apr 25 2023
%Y A291001 Cf. A000012, A015519, A033453, A289780, A291000.
%K A291001 nonn,easy
%O A291001 0,2
%A A291001 _Clark Kimberling_, Aug 22 2017