A289786 p-INVERT of the odd positive integers (A005408), where p(S) = 1 - S - S^2.

1, 5, 20, 77, 291, 1098, 4149, 15689, 59332, 224369, 848447, 3208370, 12132345, 45878109, 173486772, 656035301, 2480778763, 9380993978, 35473960589, 134143768193, 507260826084, 1918192318185, 7253589435975, 27429241169378, 103722891648049, 392225150722037

Offset: 0

Author

Clark Kimberling, Aug 10 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 A289780 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 (5, -6, 5, 1)

Crossrefs

Cf. A005408, A289780.

Programs

Mathematica

z = 60; s = x (1 + x)/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289786 *)
LinearRecurrence[{5,-6,5,1},{1,5,20,77},30] (* Harvey P. Dale, May 06 2018 *)

Formula

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

a(n) = 5*a(n-1) - 6*a(n-2) + 5*a(n-3) + a(n-4).