A291734 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = (1 - S)(1 - 2 S).

3, 7, 18, 45, 108, 258, 615, 1459, 3453, 8164, 19287, 45540, 107496, 253695, 598659, 1412587, 3332970, 7863853, 18553752, 43774722, 103279023, 243668295, 574890057, 1356344056, 3200033343, 7549859464, 17812425600, 42024945087, 99149648967, 233924207559

Offset: 0

Author

Clark Kimberling, Sep 11 2017

Keywords

• nonn
• easy

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 (3, -2, 3, -4, 0, -2)

Crossrefs

Cf. A154272, A291728.

Programs

Mathematica

z = 60; s = x + x^3; p = (1 - s) (1 - 2 s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291734 *)

Formula

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

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