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 A291728 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2. 31
 1, 2, 4, 9, 17, 35, 70, 142, 285, 576, 1160, 2340, 4716, 9510, 19171, 38653, 77926, 157110, 316747, 638599, 1287479, 2595698, 5233196, 10550681, 21271280, 42885152, 86460984, 174314476, 351436368, 708532813, 1428476905, 2879960190, 5806303628, 11706120825 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). 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: *** p(S)             t(1,0,1,0,0,0,0,...) 1 - S                A000930 (Narayana's cows sequence) 1 - S^2              A002478 (except for 0's) 1 - S^3              A291723 1 - S^5              A291724 (1 - S)^2            A291725 (1 - S)^3            A291726 (1 - S)^4            A291727 1 - S - S^2          A291728 1 - 2S - S^2         A291729 1 - 2S - 2S^2        A291730 (1 - 2S)^2           A291732 (1 - S)(1 - 2S)      A291734 1 - S - S^3          A291735 1 - S^2 - S^3        A291736 1 - S - S^2 - S^3    A291737 1 - S - S^4          A291738 1 - S^3 - S^6        A291739 (1 - S)(1 - S^2)     A291740 (1 - S)(1 + S^2)     A291741 LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 2, 0, 1) FORMULA G.f.: (-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x + x^2 + x^3 + 2 x^4 + x^6). a(n) = a(n-1) + a(n-2) + a(n-3) + 2*(a(n-4) + a(n-6) for n >= 7. MATHEMATICA z = 60; s = x + x^3; p = 1 - s - s^2; Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *) Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291728 *) CROSSREFS Cf. A154272, A290890, A291000, A291382, A291219, A291382. Sequence in context: A077931 A115451 A245122 * A268649 A316983 A136326 Adjacent sequences:  A291725 A291726 A291727 * A291729 A291730 A291731 KEYWORD nonn,easy AUTHOR Clark Kimberling, Sep 08 2017 STATUS approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)