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A290898 p-INVERT of the positive integers, where p(S) = 1 - S - S^4. 2
1, 3, 8, 22, 65, 203, 647, 2053, 6423, 19811, 60490, 183750, 557551, 1693921, 5157224, 15731043, 48041589, 146785994, 448475954, 1369853581, 4182850121, 12769287055, 38976737437, 118967979141, 363132913719, 1108463577238, 3383732698880, 10329587789993 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9, -34, 71, -89, 71, -34, 9, -1)

FORMULA

a(n) = 9*a(n-1) - 34*a(n-2) + 71*a(n-3) - 89*a(n-4) + 71*a(n-5) - 34*a(n-6) + 9*a(n-7) - a(n-8).

G.f.: (1 - x + x^2)*(1 - 5*x + 9*x^2 - 5*x^3 + x^4) / (1 - 9*x + 34*x^2 - 71*x^3 + 89*x^4 - 71*x^5 + 34*x^6 - 9*x^7 + x^8). - Colin Barker, Aug 18 2017

MATHEMATICA

z = 60; s = x/(1 - x)^2; p = 1 - s - s^4;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290898 *)

PROG

(PARI) Vec((1 - x + x^2)*(1 - 5*x + 9*x^2 - 5*x^3 + x^4) / (1 - 9*x + 34*x^2 - 71*x^3 + 89*x^4 - 71*x^5 + 34*x^6 - 9*x^7 + x^8) + O(x^40)) \\ Colin Barker, Aug 18 2017

CROSSREFS

Cf. A000027, A290890.

Sequence in context: A192681 A014138 A099324 * A117420 A003101 A064443

Adjacent sequences:  A290895 A290896 A290897 * A290899 A290900 A290901

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 17 2017

STATUS

approved

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Last modified October 16 12:00 EDT 2018. Contains 316263 sequences. (Running on oeis4.)