A291736 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S^2 - S^3.

0, 1, 1, 3, 5, 7, 16, 22, 47, 73, 137, 235, 410, 734, 1258, 2255, 3895, 6904, 12056, 21184, 37210, 65172, 114612, 200765, 352779, 618598, 1085950, 1905601, 3343713, 5868895, 10297254, 18073207, 31712887, 55655620, 97666401, 171392667, 300776956, 527817651

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 (0, 1, 1, 2, 3, 1, 3, 0, 1)

Crossrefs

Cf. A154272, A291728.

Programs

Mathematica

z = 60; s = x + x^3; p = 1 - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291736 *)

Formula

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

a(n) = a(n-2) + a(n-3) + 2*a(n-4) + 3*a(n-5) + a(n-6) + 3*a(n-7) + a(n-9) for n >= 10.