A305730 - OEIS

%I #50 Jun 25 2018 15:26:59

%S 0,0,2,8,60,440,3710,34608,355992,4004880,48948570,646121080,

%T 9163171732,138974771208,2244977073430,38485321258720,697867158824880,

%U 13346709412525728,268504389357870642,5668425997555046760,125302048367006296940,2894477317277845459160

%N a(n) is the total displacement of all letters in all permutations of n letters with no fixed points.

%H Seiichi Manyama, <a href="/A305730/b305730.txt">Table of n, a(n) for n = 0..400</a>

%F a(n) = n * (n+1) * A000166(n)/3 = 2/3 * A065087(n).

%F a(n) = n * (n+1)!/3 * Sum_{k=0..n} (-1)^k/k!.

%F a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/3) for n > 1.

%F a(n) = 2 * A000313(n+2). - _Alois P. Heinz_, Jun 22 2018

%F E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(3*(1 - x)^3). - _Ilya Gutkovskiy_, Jun 25 2018

%e n | 1 2 3 4 | the displacement of all letters | a(n)

%e --+---------+---------------------------------+------

%e 2 | 2 1     | 1 + 1 = 2                       |   2

%e 3 | 2 3 1   | 1 + 1 + 2 = 4                   |   8

%e   | 3 1 2   | 2 + 1 + 1 = 4                   |

%e 4 | 2 1 4 3 | 1 + 1 + 1 + 1 = 4               |  60

%e   | 2 3 4 1 | 1 + 1 + 1 + 3 = 6               |

%e   | 2 4 1 3 | 1 + 2 + 2 + 1 = 6               |

%e   | 3 1 4 2 | 2 + 1 + 1 + 2 = 6               |

%e   | 3 4 1 2 | 2 + 2 + 2 + 2 = 8               |

%e   | 3 4 2 1 | 2 + 2 + 1 + 3 = 8               |

%e   | 4 1 2 3 | 3 + 1 + 1 + 1 = 6               |

%e   | 4 3 1 2 | 3 + 1 + 2 + 2 = 8               |

%e   | 4 3 2 1 | 3 + 1 + 1 + 3 = 8               |

%o (PARI) {a(n) = n*(n+1)!/3*sum(k=0, n, (-1)^k/k!)}

%Y Cf. A000166, A000313, A065087, A090672.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 22 2018