A059480 a(0) = a(1) = 1; a(n) = a(n-1) + (n+1)*a(n-2).

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

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

a(n) = B(0,n)+B(1,n)+B(2,n)/2+Q(1,n)+Q(2,n)+Q(3,n)/2, n>=4, where B and Q are defined in the Mathematica section below. - Benedict W. J. Irwin, Apr 11 2017

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 *)
RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == a[n - 1] + (n + 1) a[n - 2]}, a[n], {n, 0, 24}] (* Ray Chandler, Jul 30 2015 *)
B[j_, n_] := Sum[2 n!/((n - j - 2 k)! 2^k k!), {k, 0, n/2}]
H[t_, u_, v_, n_] := HypergeometricPFQRegularized[{1, t+k-n}, {1+(u+k-n)/2, (v+k-n)/2}, -1/2]
Q[t_, n_] := Sqrt[Pi]n!Sum[((-1)^k 2^(k/2)(H[t, t, t+1, n]+(-t-k+n)H[t+1, t, t+3, n])HypergeometricU[1-k/2, 3/2, 1/2]Binomial[-t+n, k])/(n-t+1)!, {k, 1, n-t}]
Flatten[{1, 1, 4, 8, FullSimplify@Table[B[0, n] + B[1, n] + B[2, n]/2 + Q[1, n] + Q[2, n] + Q[3, n]/2, {n, 4, 20}]}] (* Benedict W. J. Irwin, Apr 11 2017 *)
nxt[{n_, a_, b_}]:={n+1, b, b+a(n+2)}; NestList[nxt, {1, 1, 1}, 30][[All, 2]] (* Harvey P. Dale, Dec 31 2017 *)

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); ) }  \\ Harry J. Smith, Jun 27 2009

Crossrefs

Sequence in context: A090083 A034515 A189546 * A105723 A280118 A143555

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