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A350134 Number of endofunctions on [n] with at least one isolated fixed point.

%I A350134 #23 Apr 27 2022 08:46:09
%S A350134 0,1,1,10,87,1046,15395,269060,5440463,124902874,3208994379,
%T A350134 91208536112,2841279322871,96258245162678,3523457725743059,
%U A350134 138573785311560916,5827414570508386335,260928229315498155314,12393729720071855683739,622422708333615857463608
%N A350134 Number of endofunctions on [n] with at least one isolated fixed point.
%H A350134 Alois P. Heinz, <a href="/A350134/b350134.txt">Table of n, a(n) for n = 0..386</a>
%F A350134 a(n) = A000312(n) - abs(A069856(n)).
%F A350134 a(n) = Sum_{k=1..n} A350212(n,k).
%e A350134 a(3) = 10: 123, 122, 133, 132, 121, 323, 321, 113, 223, 213.
%p A350134 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350134 b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
%p A350134       b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1..n))
%p A350134     end:
%p A350134 a:= n-> b(n, 0):
%p A350134 seq(a(n), n=0..23);
%t A350134 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350134 b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
%t A350134      b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1, n}]];
%t A350134 a[n_] := b[n, 0];
%t A350134 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 27 2022, after _Alois P. Heinz_ *)
%Y A350134 Column k=1 of A347999.
%Y A350134 Cf. A000312, A001865, A045531, A069856, A204042, A350212.
%K A350134 nonn
%O A350134 0,4
%A A350134 _Alois P. Heinz_, Dec 15 2021