A350134 - OEIS

%I #23 Apr 27 2022 08:46:09

%S 0,1,1,10,87,1046,15395,269060,5440463,124902874,3208994379,

%T 91208536112,2841279322871,96258245162678,3523457725743059,

%U 138573785311560916,5827414570508386335,260928229315498155314,12393729720071855683739,622422708333615857463608

%N Number of endofunctions on [n] with at least one isolated fixed point.

%H Alois P. Heinz, <a href="/A350134/b350134.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) = A000312(n) - abs(A069856(n)).

%F a(n) = Sum_{k=1..n} A350212(n,k).

%e a(3) = 10: 123, 122, 133, 132, 121, 323, 321, 113, 223, 213.

%p g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:

%p b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*

%p       b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1..n))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..23);

%t g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];

%t b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*

%t      b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1, n}]];

%t a[n_] := b[n, 0];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 27 2022, after _Alois P. Heinz_ *)

%Y Column k=1 of A347999.

%Y Cf. A000312, A001865, A045531, A069856, A204042, A350212.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Dec 15 2021