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 A290890 p-INVERT of the positive integers, where p(S) = 1 - S^2. 65
 0, 1, 4, 11, 28, 72, 188, 493, 1292, 3383, 8856, 23184, 60696, 158905, 416020, 1089155, 2851444, 7465176, 19544084, 51167077, 133957148, 350704367, 918155952, 2403763488, 6293134512, 16475640049, 43133785636, 112925716859, 295643364940, 774004377960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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). Note that in A290890, s = (1,2,3,4,...); i.e., A000027(n+1) for n>=0, whereas in A290990, s = (0,1,2,3,4,...); i.e., A000027(n) for n>=0. Guide to p-INVERT sequences using s = (1,2,3,4,5,...) = A000027: p(S)                 t(1,2,3,4,5,...) 1 - S                    A001906 1 - S^2                  A290890; see A113067 for signed version 1 - S^3                  A290891 1 - S^4                  A290892 1 - S^5                  A290893 1 - S^6                  A290894 1 - S^7                  A290895 1 - S^8                  A290896 1 - S - S^2              A289780 1 - S - S^3              A290897 1 - S - S^4              A290898 1 - S^2 - S^4            A290899 1 - S^2 - S^3            A290900 1 - S^3 - S^4            A290901 1 - 2S                   A052530;  (1/2)*A052530 = A001353 1 - 3S                   A290902;  (1/3)*A290902 = A004254 1 - 4S                   A003319;  (1/4)*A003319 = A001109 1 - 5S                   A290903;  (1/5)*A290903 = A004187 1 - 2*S^2                A290904;  (1/2)*A290904 = A909905 1 - 3*S^2                A290906;  (1/3)*A290906 = A290907 1 - 4*S^2                A290908;  (1/4)*A290908 = A099486 1 - 5*S^2                A290909;  (1/5)*A290909 = A290910 1 - 6*S^2                A290911;  (1/6)*A290911 = A290912 1 - 7*S^2                A290913;  (1/7)*A290913 = A290914 1 - 8*S^2                A290915;  (1/8)*A290915 = A290916 (1 - S)^2                A290917 (1 - S)^3                A290918 (1 - S)^4                A290919 (1 - S)^5                A290920 (1 - S)^6                A290921 1 - S - 2*S^2            A290922 1 - 2*S - 2*S^2          A290923;  (1/2)*A290923 = A290924 1 - 3*S - 2*S^2          A290925 (1 - S^2)^2              A290926 (1 - S^2)^3              A290927 (1 - S^3)^2              A290928 (1 - S)(1 - S^2)         A290929 (1 - S^2)(1 - S^4)       A290930 1 - 3 S + S^2            A291025 1 - 4 S + S^2            A291026 1 - 5 S + S^2            A291027 1 - 6 S + S^2            A291028 1 - S - S^2 - S^3        A291029 1 - S - S^2 - S^3 - S^4  A201030 1 - 3 S + 2 S^3          A291031 1 - S - S^2 - S^3 + S^4  A291032 1 - 6 S                  A291033 1 - 7 S                  A291034 1 - 8 S                  A291181 1 - 3 S + 2 S^3          A291031 1 - 3 S + 2 S^2          A291182 1 - 4 S + 2 S^3          A291183 1 - 4 S + 3 S^3          A291184 LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4, -5, 4, -1) FORMULA G.f.: x/(1 - 4 x + 5 x^2 - 4 x^3 + x^4). a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - a(n-4). EXAMPLE (See the examples at A289780.) MATHEMATICA z = 60; s = x/(1 - x)^2; p = 1 - s^2; Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *) Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290890 *) CROSSREFS Cf. A000027, A113067, A289780, A113067 (signed version of same sequence). Sequence in context: A245124 A020964 A113067 * A152689 A217918 A000604 Adjacent sequences:  A290887 A290888 A290889 * A290891 A290892 A290893 KEYWORD nonn,easy AUTHOR Clark Kimberling, Aug 15 2017 STATUS approved

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Last modified February 20 21:29 EST 2019. Contains 320350 sequences. (Running on oeis4.)