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A289807
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p-INVERT of (1,2,2,3,3,4,4,...) (A080513), where p(S) = 1 - S - S^2.
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2
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1, 4, 13, 42, 133, 424, 1348, 4291, 13653, 43449, 138261, 439979, 1400101, 4455420, 14178073, 45117606, 143573662, 456881476, 1453892534, 4626590576, 14722780217, 46850970327, 149089600359, 474434334814, 1509749422360, 4804338875098, 15288412556740
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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 A289780 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)/(1 - 3 x - 2 x^2 + 5 x^3 - x^4 - 2 x^5 + x^6).
a(n) = 3*a(n-1) + 2*a(n-2) - 5*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
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MATHEMATICA
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z = 60; s = x (1 + x - x^2)/((1 - x)^2*(1 + x)); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A080513 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289807 *)
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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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