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A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up. 8
1, 1, 2, 6, 24, 120, 720, 5039, 40305, 362682, 3626190, 39881160, 478490760, 6219298800, 87055051511, 1305598835941, 20885951018102, 354999461960226, 6388879812001704, 121367620532150280, 2426930566055020080, 50956684690331669759, 1120852238721212726609 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 17:36 EST 2022. Contains 358702 sequences. (Running on oeis4.)