A292494 p-INVERT of the odd positive integers, where p(S) = 1 - S - S^2 - S^3.

1, 5, 21, 88, 362, 1470, 5940, 23996, 97028, 392592, 1588840, 6430088, 26021472, 105301184, 426118816, 1724362608, 6977946160, 28237566352, 114268643984, 462409605552, 1871227376592, 7572272759344, 30642622403664, 124001121308400, 501793808163600

Offset: 0

Author

Clark Kimberling, Oct 04 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 A292480 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 (7, -17, 23, -12, 6, 2)

Crossrefs

Cf. A005408, A292480.

Programs

Mathematica

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292494 *)

Formula

G.f.: -(((1 + x) (1 - 3 x + 6 x^2 - 3 x^3 + 3 x^4))/(-1 + 7 x - 17 x^2 + 23 x^3 - 12 x^4 + 6 x^5 + 2 x^6)).

a(n) = 7*a(n-1) - 17*a(n-2) + 23*a(n-3) + 12*a(n-4) + 6*a(n-5) + 2*a(n-6) for n >= 7.