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

Author

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

Keywords

• nonn

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.

Examples

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

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.

Programs

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 *)

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

Extensions

Offset and example corrected by Vaclav Kotesovec, Aug 24 2014

More terms from Vaclav Kotesovec, Aug 24 2014