A289847 p-INVERT of the primes (A000040), where p(S) = 1 - S - S^2.

2, 11, 53, 253, 1205, 5740, 27336, 130200, 620129, 2953634, 14067934, 67004505, 319137367, 1520027050, 7239773429, 34482491204, 164237487721, 782250685197, 3725800625523, 17745705518523, 84521448139914, 402569240665810, 1917406730442806, 9132462688572345

Offset: 0

Author

Clark Kimberling, Aug 14 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. A000040, A030017 ("INVERT" applied to the primes), A289928.

Programs

Mathematica

z = 60; s = Sum[Prime[k] x^k, {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000040 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1](* A289847 *)