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A290927
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p-INVERT of the positive integers, where p(S) = (1 - S^2)^3.
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2
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0, 3, 12, 36, 108, 331, 1008, 3027, 8992, 26502, 77592, 225806, 653544, 1882224, 5396776, 15411399, 43847688, 124331457, 351448620, 990586686, 2784612380, 7808372811, 21845061504, 60983031772, 169897677504, 472435652577, 1311365875700, 3633925019190
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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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Index entries for linear recurrences with constant coefficients, signature (12, -63, 196, -414, 636, -731, 636, -414, 196, -63, 12, -1)
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FORMULA
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a(n) = 12*a(n-1) - 63*a(n-2) + 196*a(n-3) - 414*a(n-4) + 636*a(n-5) - 731*a(n-6) + 636*a(n-7) - 414*a(n-8) + 196*a(n-9) - 63*a(n-10) + 12*a(n-11) - a(n-12).
G.f.: x*(3 - 24*x + 81*x^2 - 156*x^3 + 193*x^4 - 156*x^5 + 81*x^6 - 24*x^7 + 3*x^8) / ((1 - 3*x + x^2)^3*(1 - x + x^2)^3). - Colin Barker, Aug 19 2017
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MATHEMATICA
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z = 60; s = x/(1 - x)^2; p = (1 - s^2)^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290927 *)
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PROG
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(PARI) concat(0, Vec(x*(3 - 24*x + 81*x^2 - 156*x^3 + 193*x^4 - 156*x^5 + 81*x^6 - 24*x^7 + 3*x^8) / ((1 - 3*x + x^2)^3*(1 - x + x^2)^3) + O(x^30))) \\ Colin Barker, Aug 19 2017
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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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