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A291730 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - 2 S - 2 S^2. 3
2, 6, 18, 56, 168, 510, 1544, 4680, 14176, 42952, 130128, 394252, 1194456, 3618840, 10963960, 33217424, 100638528, 304903688, 923764032, 2798719872, 8479257216, 25689531840, 77831351040, 235804967056, 714416256800, 2164460716896, 6557647800096 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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 A291728 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 (2, 2, 2, 4, 0, 2)

FORMULA

G.f.: -((2 (1 + x^2) (1 + x + x^3))/(-1 + 2 x + 2 x^2 + 2 x^3 + 4 x^4 + 2 x^6)).

a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) + 4*a(n-4) + 2*a(n-6) for n >= 7.

MATHEMATICA

z = 60; s = x + x^3; p = 1 - 2 s - 2 s^2;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)

u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291730 *)

u / 2  (*A291731)

CROSSREFS

Cf. A154272, A291728, A291731.

Sequence in context: A148456 A148457 A182881 * A002999 A291228 A091142

Adjacent sequences:  A291727 A291728 A291729 * A291731 A291732 A291733

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Sep 11 2017

STATUS

approved

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Last modified May 17 22:15 EDT 2021. Contains 343992 sequences. (Running on oeis4.)