A348590 - OEIS

%I #26 May 16 2022 10:00:05

%S 0,1,0,9,68,845,12474,218827,4435864,102030777,2625176150,74701061831,

%T 2329237613988,78972674630005,2892636060014050,113828236497224355,

%U 4789121681108775344,214528601554419809777,10193616586275094959534,512100888749268955942015

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

%H Alois P. Heinz, <a href="/A348590/b348590.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) mod 2 = A000035(n).

%e a(3) = 9: 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+t..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 + t, n}]];

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

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)

%Y Column k=1 of A350212.

%Y Cf. A000035, A000240, A001865, A250105.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Dec 20 2021