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A273996 Number of endofunctions on [n] whose cycle lengths are factorials.

%I A273996 #15 Jun 10 2018 16:16:57
%S A273996 1,1,4,25,218,2451,33952,560407,10750140,235118665,5775676496,
%T A273996 157448312649,4716609543736,154007821275595,5443783515005760,
%U A273996 207093963680817511,8436365861409555728,366403740283162634193,16900793597898691865920,825115046704241167668025
%N A273996 Number of endofunctions on [n] whose cycle lengths are factorials.
%H A273996 Alois P. Heinz, <a href="/A273996/b273996.txt">Table of n, a(n) for n = 0..386</a>
%p A273996 b:= proc(n) option remember; local r, f, g;
%p A273996       if n=0 then 1 else r, f, g:= $0..2;
%p A273996       while f<=n do r:= r+(f-1)!*b(n-f)*
%p A273996          binomial(n-1, f-1); f, g:= f*g, g+1
%p A273996       od; r fi
%p A273996     end:
%p A273996 a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A273996 seq(a(n), n=0..20);
%t A273996 b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {f*g, g + 1}]; r]];
%t A273996 a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
%t A273996 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A273996 Cf. A000142, A060435, A116956, A272603, A273994, A273997, A273998, A305824.
%K A273996 nonn
%O A273996 0,3
%A A273996 _Alois P. Heinz_, Jun 06 2016