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A348590 Number of endofunctions on [n] with exactly one isolated fixed point.

%I A348590 #26 May 16 2022 10:00:05
%S A348590 0,1,0,9,68,845,12474,218827,4435864,102030777,2625176150,74701061831,
%T A348590 2329237613988,78972674630005,2892636060014050,113828236497224355,
%U A348590 4789121681108775344,214528601554419809777,10193616586275094959534,512100888749268955942015
%N A348590 Number of endofunctions on [n] with exactly one isolated fixed point.
%H A348590 Alois P. Heinz, <a href="/A348590/b348590.txt">Table of n, a(n) for n = 0..386</a>
%F A348590 a(n) mod 2 = A000035(n).
%e A348590 a(3) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
%p A348590 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A348590 b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
%p A348590       b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1+t..n))
%p A348590     end:
%p A348590 a:= n-> b(n, 0):
%p A348590 seq(a(n), n=0..23);
%t A348590 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}] ;
%t A348590 b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
%t A348590      b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1 + t, n}]];
%t A348590 a[n_] := b[n, 0];
%t A348590 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)
%Y A348590 Column k=1 of A350212.
%Y A348590 Cf. A000035, A000240, A001865, A250105.
%K A348590 nonn
%O A348590 0,4
%A A348590 _Alois P. Heinz_, Dec 20 2021