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A273994 Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.

%I A273994 #15 Jun 10 2018 16:16:09
%S A273994 1,1,4,27,250,2975,43296,744913,14797036,333393345,8403026320,
%T A273994 234300271811,7161316358616,238108166195263,8556626831402560,
%U A273994 330494399041444425,13654219915946513296,600870384794864432897,28060233470995898505024,1386000542545570348128235
%N A273994 Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.
%H A273994 Alois P. Heinz, <a href="/A273994/b273994.txt">Table of n, a(n) for n = 0..386</a>
%p A273994 b:= proc(n) option remember; local r, f, g;
%p A273994       if n=0 then 1 else r, f, g:= $0..2;
%p A273994       while f<=n do r:= r+(f-1)!*b(n-f)*
%p A273994          binomial(n-1, f-1); f, g:= g, f+g
%p A273994       od; r fi
%p A273994     end:
%p A273994 a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A273994 seq(a(n), n=0..20);
%t A273994 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} = {g, f + g}]; r]];
%t A273994 a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
%t A273994 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A273994 Cf. A000045, A060435, A116956, A273001, A273996, A273997, A273998, A305824.
%K A273994 nonn
%O A273994 0,3
%A A273994 _Alois P. Heinz_, Jun 06 2016