A177484 The number of permutations having one non-overlapping occurrence of 122'1'.

0, 0, 0, 0, 6, 54, 468, 3864, 32032, 269696, 2321536, 20798448, 193509888, 1897735488, 19460711424, 211113010752, 2395487617024, 28720852065280, 359273073631232, 4735262021189376, 64904470318448640, 934415802987420672, 13945275766952386560, 217951935041766097920

Offset

0,5

Comments

The variable y is responsible for indicating if we want just one non-overlapping occurrence, and the variable x is responsible for the length of the permutation.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..480

Sergey Kitaev, Segmented partially ordered generalized patterns, Theoretical Computer Science 349(3) (2005), 420-428.

Formula

E.g.f.: (1/2 + (1/4)*tan(x)*(1 + e^(2*x) + 2*e^x*sin(x)) + (1/2)*e^x*cos(x))/(1 - y*(1 + (x - 1)*(1/2 + (1/4)*tan(x)*(1 + e^(2*x) + 2*e^x*sin(x)) + (1/2)*e^x*cos(x)))).

E.g.f.: (1/2 + 1/2*exp(x)*cos(x) + 1/4*(1 + exp(2*x) + 2*exp(x)*sin(x)) * tan(x)) * (1 + (-1 + x)*(1/2 + 1/2*exp(x)*cos(x) + 1/4*(1 + exp(2*x) + 2*exp(x)*sin(x))*tan(x))). - Vaclav Kotesovec, Aug 25 2014

a(n) ~ n! * (exp(Pi) * (Pi - 2) * cosh(Pi/4)^4 - (-1)^n * exp(-Pi) * (Pi + 2) * sinh(Pi/4)^4) * 2^(n+1) * n / Pi^(n+2). - Vaclav Kotesovec, Aug 25 2014

Example

a(4) = 6 because the only bad permutations are 1243, 1342, 1432, 2341, 2431, and 3421.

Mathematica

CoefficientList[Series[(1/2 + 1/2*E^(x)*Cos[x] + 1/4*(1 + E^(2*x) + 2*E^(x)*Sin[x])*Tan[x]) * (1 + (x-1)*(1/2 + 1/2*E^(x)*Cos[x] + 1/4*(1 + E^(2*x) + 2*E^(x)*Sin[x])*Tan[x])), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 25 2014 *)

Crossrefs

Sequence in context: A125837 A065088 A202364 * A353207 A092810 A092472

Adjacent sequences: A177481 A177482 A177483 * A177485 A177486 A177487

Keyword

nonn

Author

Signy Olafsdottir (signy06(AT)ru.is), May 09 2010, May 14 2010

Extensions

Offset and example corrected by Vaclav Kotesovec, Aug 24 2014

More terms from Vaclav Kotesovec, Aug 24 2014

Status

approved