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A290992 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

Table of n, a(n) for n=0..38.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, 0, -2, 1, 0, 1)

FORMULA

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A000027, A289780, A290990, A290991.

Sequence in context: A049876 A049795 A014251 * A265742 A098010 A088533

Adjacent sequences:  A290989 A290990 A290991 * A290993 A290994 A290995

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 21 2017

STATUS

approved

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Last modified August 25 03:09 EDT 2019. Contains 326318 sequences. (Running on oeis4.)