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%I #31 Mar 11 2021 17:30:00
%S 1,1,2,6,24,120,720,5039,40305,362682,3626190,39881160,478490760,
%T 6219298800,87055051511,1305598835941,20885951018102,354999461960226,
%U 6388879812001704,121367620532150280,2426930566055020080,50956684690331669759,1120852238721212726609
%N Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.
%H Alois P. Heinz, <a href="/A177553/b177553.txt">Table of n, a(n) for n = 0..450</a>
%H Mingjia Yang, Doron Zeilberger, <a href="https://arxiv.org/abs/1805.06077">Increasing Consecutive Patterns in Words</a>, arXiv:1805.06077 [math.CO], 2018.
%H Mingjia Yang, <a href="https://doi.org/10.7282/t3-d9z1-aw94">An experimental walk in patterns, partitions, and words</a>, Ph. D. Dissertation, Rutgers University (2020).
%F a(n)/n! ~ c * (1/r)^n, where r = 1.0001738181531504504518260962714687775785823593018886... is the root of the equation Sum_{n>=0} (r^(7*n)/(7*n)! - r^(7*n+1)/(7*n+1)!) = 0, c = 1.0010191104259450282450770594076722424772755532278.... - _Vaclav Kotesovec_, Aug 29 2014
%F E.g.f.: -(7/(2*((-cos(x*cos(3*Pi/14)))*cosh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))*cosh(x*sin(3*Pi/14))* sin(3*Pi/14) - cosh(x*sin(Pi/14))* (cos(x*cos(Pi/14))*(1 + sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))) + cos(3*Pi/14)*cosh(x*sin(3*Pi/14))* sin(x*cos(3*Pi/14)) - cosh(x*cos(Pi/7))* ((1 + cos(Pi/7))*cos(x*sin(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))) + cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(Pi/7)*cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(x*cos(Pi/14))* sinh(x*sin(Pi/14)) + cos(x*cos(Pi/14))*sin(Pi/14)* sinh(x*sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))* sinh(x*sin(Pi/14)) - cos(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))* sin(3*Pi/14)*sinh(x*sin(3*Pi/14)) + cos(3*Pi/14)*sin(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14))))). - _Vaclav Kotesovec_, Jan 31 2015
%F In closed form, c = 7 / (r * (2*cos(r*sin(Pi/7))*cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + 2*cos(r*cos(Pi/14)) * cosh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * cosh(r*sin(3*Pi/14)) + 2*cosh(r*sin(Pi/14)) * sin(Pi/14 + r*cos(Pi/14)) - 2*cosh(r*sin(3*Pi/14)) * sin(3*Pi/14 - r*cos(3*Pi/14)) - 2*cos(r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - 2*cos(r*cos(Pi/14)) * sinh(r*sin(Pi/14)) - 2*sin(Pi/14 + r*cos(Pi/14))*sinh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)) - 2*sin((3*Pi)/14 - r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)))). - _Vaclav Kotesovec_, Feb 01 2015
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p `if`(t<5, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
%p add(b(u-j, o+j-1, 0), j=1..u))
%p end:
%p a:= n-> b(n, 0, 0):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 07 2013
%t nn=20;r=6;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* _Geoffrey Critzer_, Feb 25 2014 *)
%t Table[n!*SeriesCoefficient[1/(Sum[x^(7*k)/(7*k)!-x^(7*k+1)/(7*k+1)!,{k,0,n}]),{x,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Aug 29 2014 *)
%Y Column k=63 of A242784.
%Y Cf. A080635, A049774, A117158, A177533, A177523.
%K nonn
%O 0,3
%A _R. H. Hardin_, May 10 2010
%E a(18)-a(22) from _Alois P. Heinz_, Oct 07 2013
%E a(0)=1 prepended by _Alois P. Heinz_, Aug 08 2018
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