A289924 p-INVERT of (n!), n >= 1 (A000142, shifted), where p(S) = 1 - S - S^2.

1, 4, 17, 79, 402, 2253, 14037, 98152, 774973, 6911131, 69225314, 771593257, 9470565513, 126755983488, 1834510979193, 28511931874423, 473179672441090, 8346048191981797, 155838573499885229, 3069991622444141848, 63618933765102190149, 1383222300396890185731

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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..448
• Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.

Crossrefs

Cf. A000142.

Programs

Mathematica

z = 60; s = Sum[k! x^k, {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000142 shifted *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289924 *)