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A290906
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p-INVERT of the positive integers, where p(S) = 1 - 3*S^2.
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3
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0, 3, 12, 39, 132, 456, 1572, 5409, 18612, 64053, 220440, 758640, 2610840, 8985147, 30922188, 106418031, 366235308, 1260390744, 4337606988, 14927778921, 51373622388, 176801189997, 608457401520, 2093992746720, 7206429919920, 24800769855603, 85351303248012
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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 A290890 for a guide to related sequences.
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LINKS
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FORMULA
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G.f.: (3 x)/(1 - 4 x + 3 x^2 - 4 x^3 + x^4).
a(n) = 4*a(n-1) - 3*a(n-2) + 4*a(n-3) - a(n-4).
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MATHEMATICA
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z = 60; s = x/(1 - x)^2; p = 1 - 3 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290906 *)
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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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