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A059480
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a(0) = a(1) = 1; a(n) = a(n-1) + (n+1)*a(n-2).
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3
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,200
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns.
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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]
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CROSSREFS
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Sequence in context: A090083 A034515 A189546 * A105723 A143555 A025234
Adjacent sequences: A059477 A059478 A059479 * A059481 A059482 A059483
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KEYWORD
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nonn
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AUTHOR
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Wouter Meeussen, Feb 15 2001
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EXTENSIONS
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Name changed and offset of interpretation corrected by Olivier Gérard, Sep 26 2012
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STATUS
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approved
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