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A291728 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2.

%I A291728 #15 Nov 23 2024 08:28:53
%S A291728 1,2,4,9,17,35,70,142,285,576,1160,2340,4716,9510,19171,38653,77926,
%T A291728 157110,316747,638599,1287479,2595698,5233196,10550681,21271280,
%U A291728 42885152,86460984,174314476,351436368,708532813,1428476905,2879960190,5806303628,11706120825
%N A291728 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2.
%C A291728 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 A291728 In the following guide to p-INVERT sequences using s = (1,0,1,0,0,0,0,...) = A154272, in some cases t(1,0,1,0,0,0,0,...) is a shifted (or differently indexed) version of the indicated sequence:
%C A291728 ***
%C A291728 p(S)             t(1,0,1,0,0,0,0,...)
%C A291728 1 - S                A000930 (Narayana's cows sequence)
%C A291728 1 - S^2              A002478 (except for 0's)
%C A291728 1 - S^3              A291723
%C A291728 1 - S^5              A291724
%C A291728 (1 - S)^2            A291725
%C A291728 (1 - S)^3            A291726
%C A291728 (1 - S)^4            A291727
%C A291728 1 - S - S^2          A291728
%C A291728 1 - 2S - S^2         A291729
%C A291728 1 - 2S - 2S^2        A291730
%C A291728 (1 - 2S)^2           A291732
%C A291728 (1 - S)(1 - 2S)      A291734
%C A291728 1 - S - S^3          A291735
%C A291728 1 - S^2 - S^3        A291736
%C A291728 1 - S - S^2 - S^3    A291737
%C A291728 1 - S - S^4          A291738
%C A291728 1 - S^3 - S^6        A291739
%C A291728 (1 - S)(1 - S^2)     A291740
%C A291728 (1 - S)(1 + S^2)     A291741
%H A291728 Clark Kimberling, <a href="/A291728/b291728.txt">Table of n, a(n) for n = 0..1000</a>
%H A291728 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,2,0,1).
%F A291728 G.f.: (-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x + x^2 + x^3 + 2 x^4 + x^6).
%F A291728 a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-6) for n >= 7.
%t A291728 z = 60; s = x + x^3; p = 1 - s - s^2;
%t A291728 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
%t A291728 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291728 *)
%Y A291728 Cf. A154272, A290890, A291000, A291382, A291219, A291382.
%K A291728 nonn,easy
%O A291728 0,2
%A A291728 _Clark Kimberling_, Sep 08 2017