A291034 p-INVERT of the positive integers, where p(S) = 1 - 7*S.

7, 63, 560, 4977, 44233, 393120, 3493847, 31051503, 275969680, 2452675617, 21798110873, 193730322240, 1721774789287, 15302242781343, 135998410242800, 1208683449403857, 10742152634391913, 95470690260123360, 848494059706718327, 7540975847100341583

Offset: 0

Author

Clark Kimberling, Aug 19 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 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 (9, -1)

Crossrefs

Cf. A000027, A290890.

Programs

Mathematica

z = 60; s = x/(1 - x)^2; p = 1 - 7 s;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291034 *)

Formula

G.f.: 7/(1 - 9 x + x^2).

a(n) = 9*a(n-1) - a(n-2).

a(n) = 7*A018913(n) for n >= 1.