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A290990 p-INVERT of the nonnegative integers (A000027), where p(S) = 1 - S - S^2. 6
0, 1, 2, 5, 12, 28, 64, 145, 328, 743, 1686, 3830, 8704, 19781, 44950, 102133, 232048, 527208, 1197808, 2721421, 6183108, 14048151, 31917714, 72517738, 164761792, 374342057, 850512458, 1932380869, 4390407092, 9975090996, 22663602720, 51492150953 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 (4, -5, 2, 1)

FORMULA

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) + a(n-4).

G.f.: x*(1 - 2*x + 2*x^2) / (1 - 4*x + 5*x^2 - 2*x^3 - x^4). - Colin Barker, Aug 24 2017

MATHEMATICA

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

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

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290990 *)

PROG

(PARI) concat(0, Vec(x*(1 - 2*x + 2*x^2) / (1 - 4*x + 5*x^2 - 2*x^3 - x^4) + O(x^50))) \\ Colin Barker, Aug 24 2017

CROSSREFS

Cf. A000027, A289780, A290991.

Sequence in context: A019301 A006980 A045623 * A324586 A001410 A258898

Adjacent sequences:  A290987 A290988 A290989 * A290991 A290992 A290993

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 21 2017

STATUS

approved

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Last modified March 26 10:57 EDT 2019. Contains 321491 sequences. (Running on oeis4.)