cp's OEIS Frontend

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

%I A289924 #14 Mar 20 2021 11:40:39
%S A289924 1,4,17,79,402,2253,14037,98152,774973,6911131,69225314,771593257,
%T A289924 9470565513,126755983488,1834510979193,28511931874423,473179672441090,
%U A289924 8346048191981797,155838573499885229,3069991622444141848,63618933765102190149,1383222300396890185731
%N A289924 p-INVERT of (n!), n >= 1 (A000142, shifted), where p(S) = 1 - S - S^2.
%C A289924 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).
%C A289924 See A289780 for a guide to related sequences.
%H A289924 Michael De Vlieger, <a href="/A289924/b289924.txt">Table of n, a(n) for n = 0..448</a>
%H A289924 Mikhail Khovanov, Victor Ostrik and Yakov Kononov, <a href="https://arxiv.org/abs/2011.14758">Two-dimensional topological theories, rational functions and their tensor envelopes</a>, arXiv:2011.14758 [math.QA], 2020.
%t A289924 z = 60; s = Sum[k! x^k, {k, 1, z}]; p = 1 - s - s^2;
%t A289924 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000142 shifted *)
%t A289924 Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289924 *)
%Y A289924 Cf. A000142.
%K A289924 nonn,easy
%O A289924 0,2
%A A289924 _Clark Kimberling_, Aug 14 2017