login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A177523 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up. 35
1, 1, 2, 6, 24, 119, 709, 4928, 39144, 349776, 3472811, 37928331, 451891992, 5832672456, 81074690424, 1207441809209, 19181203110129, 323753459184738, 5785975294622694, 109149016813544376, 2167402030585724571, 45190632809497874161, 987099099863360190632 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of permutations of length n that avoid the consecutive pattern 12345 (or equivalently 54321).

LINKS

Ray Chandler and Alois P. Heinz, Table of n, a(n) for n = 0..400 (terms n = 1..40 from Ray Chandler)

A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents , 2014. See displayed equation before Eq. (3), and set m=5. - N. J. A. Sloane, Aug 11 2014

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

FORMULA

E.g.f.: 1/( Sum_{n>=0} x^(5*n)/(5*n)! - x^(5*n+1)/(5*n+1)! ).

a(n)/n! ~ c * (1/r)^n, where r = 1.007187547786015395418998654... is the root of the equation Sum_{n>=0} (r^(5*n)/(5*n)! - r^(5*n+1)/(5*n+1)!) = 0, c = 1.02806793756750152.... - Vaclav Kotesovec, Dec 11 2013

Equivalently, r = 1.00718754778601539541899865400272701484... is the root of the equation (5+sqrt(5)) * cos(sqrt((5-sqrt(5))/2)*r/2) + (5-sqrt(5)) * exp(sqrt(5)*r/2) * cos(sqrt((5+sqrt(5))/2)*r/2) - sqrt(2*(5-sqrt(5))) * sin(sqrt((5-sqrt(5))/2)*r/2) - sqrt(2*(5+sqrt(5))) * exp(sqrt(5)*r/2) * sin(sqrt((5+sqrt(5))/2)*r/2) = 0. - Vaclav Kotesovec, Aug 29 2014

E.g.f.: 10*exp((1+sqrt(5))*x/4) / ((5+sqrt(5)) * cos(sqrt((5-sqrt(5))/2)*x/2) + (5-sqrt(5)) * exp(sqrt(5)*x/2) * cos(sqrt((5+sqrt(5))/2)*x/2) - sqrt(2*(5-sqrt(5))) * sin(sqrt((5-sqrt(5))/2)*x/2) - sqrt(2*(5+sqrt(5))) * exp(sqrt(5)*x/2) * sin(sqrt((5+sqrt(5))/2)*x/2)). - Vaclav Kotesovec, Aug 29 2014

In closed form, c = 5*exp((1+sqrt(5))*r/4) / (r*((5 + sqrt(5)) * cos(sqrt((5 - sqrt(5))/2)*r/2) + (5 - sqrt(5)) * exp(sqrt(5)*r/2) * cos(sqrt((5 + sqrt(5))/2)*r/2))) = 1.0280679375675015201596831656779442465978511664638... . Vaclav Kotesovec, Feb 01 2015

EXAMPLE

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 24*x^4/4! + 119*x^5/5! + 709*x^6/6! +...

where A(x) = 1/(1 - x + x^5/5! - x^6/6! + x^10/10! - x^11/11! + x^15/15! - x^16/16! + x^20/20! +...).

MATHEMATICA

Table[n!*SeriesCoefficient[1/(Sum[x^(5*k)/(5*k)!-x^(5*k+1)/(5*k+1)!, {k, 0, n}]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 11 2013 *)

FullSimplify[CoefficientList[Series[10*E^((1+Sqrt[5])*x/4) / ((5+Sqrt[5]) * Cos[Sqrt[(5-Sqrt[5])/2]*x/2] + (5-Sqrt[5]) * E^(Sqrt[5]*x/2) * Cos[Sqrt[(5+Sqrt[5])/2]*x/2] - Sqrt[2*(5-Sqrt[5])] * Sin[Sqrt[(5-Sqrt[5])/2]*x/2] - Sqrt[2*(5+Sqrt[5])] * E^(Sqrt[5]*x/2) * Sin[Sqrt[(5+Sqrt[5])/2]*x/2]), {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Aug 29 2014 *)

PROG

(PARI) {a(n)=n!*polcoeff(1/sum(m=0, n\5+1, x^(5*m)/(5*m)!-x^(5*m+1)/(5*m+1)!+x^2*O(x^n)), n)}

CROSSREFS

Cf. A080635, A049774, A117158, A177533.

Column k=15 of A242784.

Sequence in context: A202233 A202235 A202236 * A005395 A092495 A110808

Adjacent sequences:  A177520 A177521 A177522 * A177524 A177525 A177526

KEYWORD

nonn

AUTHOR

R. H. Hardin, May 10 2010

EXTENSIONS

More terms from Ray Chandler, Dec 06 2011

a(0)=1 prepended by Alois P. Heinz, Jan 13 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 23 00:56 EST 2019. Contains 319365 sequences. (Running on oeis4.)