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 A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3. 57
 1, 3, 9, 26, 74, 210, 596, 1692, 4804, 13640, 38728, 109960, 312208, 886448, 2516880, 7146144, 20289952, 57608992, 163568448, 464417728, 1318615104, 3743926400, 10630080640, 30181847168, 85694918912, 243312448256, 690833811712, 1961475291648, 5569190816256 (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 = (1,1,1,1,1,...) = A000012, in some cases t(1,1,1,1,1,...) is a shifted version of the cited sequence: p(S)                             t(1,1,1,1,1,...) 1 - S                                A000079 1 - S^2                              A000079 1 - S^3                              A024495 1 - S^4                              A000749 1 - S^5                              A139761 1 - S^6                              A290993 1 - S^7                              A290994 1 - S^8                              A290995 1 - S - S^2                          A001906 1 - S - S^3                          A116703 1 - S - S^4                          A290996 1 - S^3 - S^6                        A290997 1 - S^2 - S^3                        A095263 1 - S^3 - S^4                        A290998 1 - 2 S^2                            A052542 1 - 3 S^2                            A002605 1 - 4 S^2                            A015518 1 - 5 S^2                            A163305 1 - 6 S^2                            A290999 1 - 7 S^2                            A291008 1 - 8 S^2                            A291001 (1 - S)^2                            A045623 (1 - S)^3                            A058396 (1 - S)^4                            A062109 (1 - S)^5                            A169792 (1 - S)^6                            A169793 (1 - S^2)^2                          A024007 1 - 2 S - 2 S^2                      A052530 1 - 3 S - 2 S^2                      A060801 (1 - S)(1 - 2 S)                     A053581 (1 - 2 S)(1 - 3 S)                   A291002 (1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S)   A291003 (1 - 2 S)^2                          A120926 (1 - 3 S)^2                          A291004 1 + S - S^2                          A000045  (Fibonacci numbers starting with -1) 1 - S - S^2 - S^3                    A291000 1 - S - S^2 - S^3 - S^4              A291006 1 - S - S^2 - S^3 - S^4 - S^5        A291007 1 - S^2 - S^4                        A290990 (1 - S)(1 - 3 S)                     A291009 (1 - S)(1 - 2 S)(1 - 3 S)            A291010 (1 - S)^2 (1 - 2 S)                  A291011 (1 - S^2)(1 - 2 S)                   A291012 (1 - S^2)^3                          A291013 (1 - S^3)^2                          A291014 1 - S - S^2 + S^3                    A045891 1 - 2 S - S^2 + S^3                  A291015 1 - 3 S + S^2                        A136775 1 - 4 S + S^2                        A291016 1 - 5 S + S^2                        A291017 1 - 6 S + S^2                        A291018 1 - S - S^2 - S^3 + S^4              A291019 1 - S - S^2 - S^3 - S^4 + S^5        A291020 1 - S - S^2 - S^3 + S^4 + S^5        A291021 1 - S - 2 S^2 + 2 S^3                A175658 1 - 3 S^2 + 2 S^3                    A291023 (1 - 2 S^2)^2                        A291024 (1 - S^3)^3                          A291143 (1 - S - S^2)^2                      A209917 LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4, -4, 2) FORMULA G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3). a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) for n >= 4. MATHEMATICA z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3; Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *) Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291000 *) CROSSREFS Cf. A000012, A289780. Sequence in context: A127911 A116423 A077845 * A276068 A171277 A289806 Adjacent sequences:  A290997 A290998 A290999 * A291001 A291002 A291003 KEYWORD nonn,easy AUTHOR Clark Kimberling, Aug 22 2017 STATUS approved

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