A291248 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4 + S^5.

1, 2, 5, 12, 27, 65, 146, 346, 788, 1845, 4239, 9865, 22758, 52818, 122072, 282954, 654528, 1516221, 3508817, 8125763, 18808494, 43550500, 100815652, 233418699, 540371471, 1251079052, 2896357943, 6705591388, 15524220275, 35941069252, 83208225215

Offset: 0

Author

Clark Kimberling, Aug 29 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 A291219 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 (1, 6, -3, -12, 3, 12, -3, -6, 1, 1)

Crossrefs

Cf. A000035, A291219.

Programs

Mathematica

z = 60; s = x/(1 - x^2); p = 1 - s - s^2 - s^3 - s^4 + s^5;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291248 *)

Formula

G.f.: -((1 + x - 3 x^2 - 2 x^3 + 3 x^4 + 2 x^5 - 3 x^6 - x^7 + x^8)/((-1 - x + x^2) (1 - 2 x - 3 x^2 + 4 x^3 + 5 x^4 - 4 x^5 - 3 x^6 + 2 x^7 + x^8))).

a(n) = a(n-1) + 6*a(n-2) - 3*a(n-3) - 12*a(n-4) + 3*a(n-5) + 12*a(n-6) - 3*a(n-7) - 6*a(n-8) + a(n-9) + a(n-10) for n >= 11.