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A059480 a(0) = a(1) = 1; a(n) = a(n-1) + (n+1)*a(n-2). 3
1, 1, 4, 8, 28, 76, 272, 880, 3328, 12128, 48736, 194272, 827840, 3547648, 15965248, 72727616, 344136832, 1653233920, 8191833728, 41256512128, 213285020416, 1120928287232, 6026483756800, 32928762650368, 183590856570368 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of permutations of order (n+4) that simultaneously avoid the patterns 12-3 and 21-3, start with 1 and end with pattern 12.

LINKS

Harry J. Smith, Table of n, a(n) for n=0,...,200

S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns.

FORMULA

a(n) = a(n - 1) + (n + 1)*a(n - 2); a(0) = a(1) = 1;

E.g.f.: (-2*(1+x)+ e^((x*(2+x))/2)*(2+x*(2+x))*(2 +sqrt(2*e*Pi) * erf(1/sqrt(2))) - e^((1+x)^2/2)*sqrt(2*Pi)*(2+x*(2+x)) * erf((1+x)/sqrt(2)))/2.

E.g.f.: (with offset 2) exp(x+x^2/2) * (1-integral(exp(-t-t^2/2) dt, t=0..x)) - 1 .

a(n) ~ (1/sqrt(2) + sqrt(Pi)/2*exp(1/2) * (erf(1/sqrt(2)) - 1)) * n^(n/2+1)*exp(sqrt(n) - n/2 - 1/4) * (1+31/(24*sqrt(n))). - Vaclav Kotesovec, Dec 27 2012

EXAMPLE

For n=3, the a(3) = 8 permutations of n+4=7 symbols that satisfy the constraints are: {1,7,2,6,5,3,4},{1,7,3,6,5,2,4},{1,7,4,6,5,2,3},{1,7,5,6,4,2,3},{1,7,6,2,5,3,4},{1,7,6,3,5,2,4},{1,7,6,4,5,2,3} and {1,7,6,5,4,2,3}. - Olivier Gérard, Sep 26 2012

MATHEMATICA

FullSimplify[CoefficientList[Series[1/2*((Sqrt[2*E*Pi]*Erf[1/Sqrt[2]]+2) * E^(1/2*x*(x+2))*(x*(x+2)+2)-Sqrt[2*Pi]*E^(1/2*(x+1)^2)*(x*(x+2)+2) * Erf[(x+1)/Sqrt[2]]-2*(x+1)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 27 2012 *)

Rest@ LinearRecurrence[{1, 1 + #}, {0, 1}, 23] (* Robert G. Wilson v, Jul 12 2014 *)

PROG

(PARI) { a=b=c=1; for (n = 0, 200, if (n>1, a=b + (n + 1)*c; c=b; b=a); write("b059480.txt", n, " ", a); ) } [From Harry J. Smith, Jun 27 2009]

CROSSREFS

Sequence in context: A090083 A034515 A189546 * A105723 A143555 A025234

Adjacent sequences:  A059477 A059478 A059479 * A059481 A059482 A059483

KEYWORD

nonn

AUTHOR

Wouter Meeussen, Feb 15 2001

EXTENSIONS

Name changed and offset of interpretation corrected by Olivier Gérard, Sep 26 2012

STATUS

approved

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Last modified October 30 13:32 EDT 2014. Contains 248803 sequences.