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A291028 p-INVERT of the positive integers, where p(S) = 1 - 6*S + S^2. 2
6, 47, 362, 2787, 21456, 165180, 1271644, 9789793, 75367038, 580215573, 4466808294, 34387867640, 264736107506, 2038079457267, 15690220398162, 120791667500967, 929918545909756, 7159007901103540, 55113853093361544, 424295774604244773, 3266454697733704038 (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 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 (10,-19,10,-1).

FORMULA

G.f.: (6 - 13 x + 6 x^2)/(1 - 10 x + 19 x^2 - 10 x^3 + x^4).

a(n) = 10*a(n-1) - 19*a(n-2) + 10*a(n-3) - a(n-4).

MATHEMATICA

z = 60; s = x/(1 - x)^2; p = 1 - 6 s + s^2;

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

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291028 *)

LinearRecurrence[{10, -19, 10, -1}, {6, 47, 362, 2787}, 40] (* Vincenzo Librandi, Aug 20 2017 *)

PROG

(MAGMA) I:=[6, 47, 362, 2787]; [n le 4 select I[n] else 10*Self(n-1)-19*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 20 2017

CROSSREFS

Cf. A000027, A290890.

Sequence in context: A267203 A024076 A015553 * A071878 A104256 A289211

Adjacent sequences:  A291025 A291026 A291027 * A291029 A291030 A291031

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 19 2017

STATUS

approved

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Last modified January 22 19:39 EST 2020. Contains 331153 sequences. (Running on oeis4.)