A289928 p-INVERT of (1,2,3,5,7,11,13,...); i.e., 1 and the primes (A008578), where p(S) = 1 - S - S^2.

1, 4, 14, 48, 162, 547, 1842, 6206, 20906, 70438, 237326, 799629, 2694199, 9077599, 30585239, 103051135, 347211149, 1169861760, 3941626163, 13280557904, 44746308037, 150764154490, 507971076799, 1711511703373, 5766612400708, 19429501132982, 65464000013233

Offset: 0

Author

Clark Kimberling, Aug 15 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.

Crossrefs

Cf. A008578, A289847.

Programs

Mathematica

z = 60; s = x + Sum[Prime[k] x^(k + 1), {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A008578 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1] (* A289928 *)