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A291239
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p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S^2) (1 - 2 S).
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2
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2, 5, 12, 31, 74, 184, 442, 1081, 2604, 6323, 15250, 36912, 89074, 215293, 519660, 1255223, 3030106, 7317032, 17664170, 42649553, 102963276, 248587051, 600137378, 1448890464, 3497918306, 8444802101, 20387522508, 49220043535, 118827609578, 286875776920
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OFFSET
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0,1
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COMMENTS
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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).
See A291219 for a guide to related sequences.
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LINKS
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FORMULA
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G.f.: (-2 - x + 6 x^2 + x^3 - 2 x^4)/(-1 + 2 x + 4 x^2 - 6 x^3 - 4 x^4 + 2 x^5 + x^6).
a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4) + 2*a(n-5) + a(n-6) for n >= 7.
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MATHEMATICA
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z = 60; s = x/(1 - x^2); p = (1 - s^2)(1 - 2s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291239 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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