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A290992
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p-INVERT of (0,0,0,1,2,3,4,5,...), the nonnegative integers A000027 preceded by two zeros, where p(S) = 1 - S - S^2.
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3
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0, 0, 0, 1, 2, 3, 4, 7, 14, 27, 48, 82, 140, 242, 420, 726, 1250, 2153, 3720, 6446, 11184, 19408, 33676, 58431, 101378, 175861, 304988, 528800, 916714, 1589091, 2754612, 4775074, 8277754, 14350253, 24878304, 43131381, 74777890, 129645147, 224770632
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OFFSET
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0,5
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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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a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-5) + a(n-6) + a(n-8).
G.f.: x^3*(1 - 2*x + x^2 + x^4) / (1 - 4*x + 6*x^2 - 4*x^3 + 2*x^5 - x^6 - x^8). - Colin Barker, Aug 24 2017
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MATHEMATICA
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z = 60; s = x^4/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* 0, 0, 0, 1, 2, 3, 4, 5, ... *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290992 *)
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PROG
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(PARI) concat(vector(3), Vec(x^3*(1 - 2*x + x^2 + x^4) / (1 - 4*x + 6*x^2 - 4*x^3 + 2*x^5 - x^6 - x^8) + O(x^50))) \\ Colin Barker, Aug 24 2017
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0, 0, 0] cat Coefficients(R!( x^3*(1-2*x+x^2+x^4)/(1-4*x+6*x^2-4*x^3+2*x^5-x^6-x^8) )); // G. C. Greubel, Apr 12 2023
(SageMath)
def f(x): return x^3*(1-2*x+x^2+x^4)/(1-4*x+6*x^2-4*x^3+2*x^5-x^6-x^8)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(x) ).list()
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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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