A292478 p-INVERT of A010892, where p(S) = 1 - S - S^3.

1, 2, 4, 9, 19, 36, 66, 127, 256, 513, 998, 1916, 3717, 7311, 14396, 28134, 54719, 106624, 208577, 408426, 798340, 1558049, 3041323, 5942916, 11618026, 22703871, 44346624, 86617729, 169226638, 330675804, 646109597, 1262265767, 2465931324, 4817687118

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).

Links

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

Index entries for linear recurrences with constant coefficients, signature (4, -8, 11, -8, 4, -1)

Formula

G.f.: (1 - 2 x + 4 x^2 - 2 x^3 + x^4)/(1 - 4 x + 8 x^2 - 11 x^3 + 8 x^4 - 4 x^5 + x^6).

a(n) = 4*a(n-1) - 8*a(n-2) + 11*a(n-3) - 8*a(n-4) + 4*a(n-5) - a(n-6) for n >= 7.

Mathematica

z = 60; s = x/(1 - x + x^2); p = 1 - s - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A010892 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292478 *)
LinearRecurrence[{4, -8, 11, -8, 4, -1}, {1, 2, 4, 9, 19, 36}, 40] (* Vincenzo Librandi, Oct 01 2017 *)

Prog

(Magma) I:=[1, 2, 4, 9, 19, 36]; [n le 6 select I[n] else 4*Self(n-1)-8*Self(n-2)+11*Self(n-3)-8*Self(n-4)+4*Self(n-5)-Self(n-6): n in [1..35]]; // Vincenzo Librandi, Oct 01 2017

Crossrefs

Cf. A010892.

Sequence in context: A331850 A373808 A081490 * A309267 A262864 A129784

Adjacent sequences: A292475 A292476 A292477 * A292479 A292480 A292481

Keyword

nonn,easy

Author

Clark Kimberling, Sep 30 2017

Status

approved