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A292322
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p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S - S^2 - S^3.
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4
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1, 2, 4, 8, 17, 36, 73, 152, 317, 653, 1355, 2812, 5818, 12061, 25001, 51786, 107323, 222409, 460824, 954942, 1978840, 4100398, 8496827, 17606974, 36484494, 75602461, 156661630, 324629762, 672690133, 1393931744, 2888469094, 5985414154, 12402824741
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OFFSET
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0,2
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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 A291728 for a guide to related sequences.
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LINKS
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FORMULA
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G.f.: -((1 + x + x^2 - 2 x^3 - x^4 + x^6)/(-1 + x + x^2 + 4 x^3 - 2 x^4 - x^5 - 3 x^6 + x^7 + x^9)).
a(n) = a(n-1) + a(n-2) + 4*a(n-3) - 2*a(n-4) - a(n-5) - 3*a(n-6) + a(n-7) + a(n-9) for n >= 10.
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MATHEMATICA
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z = 60; s = x/(x - x^3); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A079978 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292322 *)
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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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