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A291736
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p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S^2 - S^3.
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2
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0, 1, 1, 3, 5, 7, 16, 22, 47, 73, 137, 235, 410, 734, 1258, 2255, 3895, 6904, 12056, 21184, 37210, 65172, 114612, 200765, 352779, 618598, 1085950, 1905601, 3343713, 5868895, 10297254, 18073207, 31712887, 55655620, 97666401, 171392667, 300776956, 527817651
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OFFSET
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0,4
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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.: -((x (1 + x^2)^2 (1 + x + x^3))/((-1 + 2 x - x^2 + x^3) (1 + 2 x + 2 x^2 + 2 x^3 + 2 x^4 + x^5 + x^6))).
a(n) = a(n-2) + a(n-3) + 2*a(n-4) + 3*a(n-5) + a(n-6) + 3*a(n-7) + a(n-9) for n >= 10.
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MATHEMATICA
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z = 60; s = x + x^3; p = 1 - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291736 *)
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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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