A093965 Number of functions of [n] to [n] that simultaneously avoid the patterns 112 and 221.

1, 4, 21, 124, 825, 6186, 51961, 484968, 4988241, 56117710, 685883121, 9053657196, 128397320233, 1947359356866, 31457343457065, 539268744978256, 9778739908939041, 187018400758459158, 3762370179964296001, 79427814910357360020, 1755772750650004800441

Offset: 1

Author

Ralf Stephan, Apr 20 2004

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A093966.

Programs

Magma

[n le 2 select 4^(n-1) else n*((n+1)*Self(n-1) - (n-1)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Dec 29 2021

Mathematica

Rest[CoefficientList[Series[x(E^x-x)/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Nov 20 2012 *)

PARI

my(x='x+O('x^66)); Vec(serlaplace(x*(exp(x)-x)/(1-x)^2)) \\ Joerg Arndt, May 11 2013

Sage

[factorial(n)*( x*(exp(x) -x)/(1-x)^2 ).series(x,n+1).list()[n] for n in (1..30)] # G. C. Greubel, Dec 29 2021

Formula

a(n) = A093966(n, n).

From Vaclav Kotesovec, Nov 20 2012: (Start)

E.g.f.: x*(exp(x) - x)/(1-x)^2.

Recurrence: (n-1)*a(n) = n*(n+1)*a(n-1) - (n-1)*n*a(n-2) for n>2.

a(n) ~ n!*n*(e-1). (End)

Extensions

Name changed by Olivier Gérard, Aug 06 2016