%I #25 Mar 29 2023 15:25:10
%S 1,1,1,1,1,1,2,14,98,546,2562,10626,41118,174174,1093092,10005996,
%T 98041944,889104216,7315812504,55893493656,421564046904,3519008733240,
%U 36011379484080,435775334314320,5538098453968080,68428271204813520,805379194188288720
%N Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 6).
%H Seiichi Manyama, <a href="/A322262/b322262.txt">Table of n, a(n) for n = 0..514</a>
%H David Galvin, John Engbers, and Clifford Smyth, <a href="https://arxiv.org/abs/2303.14057">Reciprocals of thinned exponential series</a>, arXiv:2303.14057 [math.CO], 2023.
%H Ira M. Gessel, <a href="https://arxiv.org/abs/1807.09290">Reciprocals of exponential polynomials and permutation enumeration</a>, arXiv:1807.09290 [math.CO], 2018.
%F E.g.f.: 1/(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5!).
%e For n=6 the a(6)=2 permutations are 654321 and 123456.
%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/sum(k=0, 5, (-x)^k/k!)))
%Y Cf. A000142, A322251 (mod 3), A317111 (mod 4), A322276 (mod 5).
%K nonn
%O 0,7
%A _Seiichi Manyama_, Dec 01 2018