A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

1, 1, 2, 6, 24, 120, 720, 5039, 40305, 362682, 3626190, 39881160, 478490760, 6219298800, 87055051511, 1305598835941, 20885951018102, 354999461960226, 6388879812001704, 121367620532150280, 2426930566055020080, 50956684690331669759, 1120852238721212726609

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Formula

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

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

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

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<5, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013

Mathematica

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 *)
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 *)

Crossrefs

Column k=63 of A242784.

Cf. A080635, A049774, A117158, A177533, A177523.

Sequence in context: A226438 A248839 A052399 * A090583 A248775 A108889

Adjacent sequences: A177550 A177551 A177552 * A177554 A177555 A177556

Keyword

nonn

Author

R. H. Hardin, May 10 2010

Extensions

a(18)-a(22) from Alois P. Heinz, Oct 07 2013

a(0)=1 prepended by Alois P. Heinz, Aug 08 2018

Status

approved