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A231812 Number of endofunctions on [n] where all nonempty preimages have the same cardinality.

%I A231812 #21 Oct 15 2018 09:26:18
%S A231812 1,1,4,9,64,125,2826,5047,218688,504009,32216950,39916811,7585223196,
%T A231812 6227020813,2424646536326,1813027195995,1072898135852416,
%U A231812 355687428096017,616925243565037854,121645100408832019,441395941479128984940,72313131901887676821
%N A231812 Number of endofunctions on [n] where all nonempty preimages have the same cardinality.
%C A231812 Number of endofunctions f:{1,...,n}-> {1,...,n} such that (1<=i<j<=n and |f^(-1)(i)|>0 and |f^(-1)(j)|>0) implies |f^(-1)(i)| = |f^(-1)(j)|.
%H A231812 Alois P. Heinz, <a href="/A231812/b231812.txt">Table of n, a(n) for n = 0..300</a>
%F A231812 a(n) = Sum_{d|n} multinomial(n; {n/d}^d)*C(n,d) for n>0, a(0) = 1.
%F A231812 a(n) = n! + n = A005095(n) for prime n.
%e A231812 a(2) = 4: (1,1), (1,2), (2,1), (2,2).
%e A231812 a(3) = 9: (1,1,1), (1,2,3), (1,3,2), (2,1,3), (2,2,2), (2,3,1), (3,1,2), (3,2,1), (3,3,3).
%e A231812 a(4) = 64: (1,1,1,1), (1,1,2,2), (1,1,3,3), ..., (4,4,3,3), (4,4,4,4).
%p A231812 with(numtheory): with(combinat): C:= binomial:
%p A231812 a:= n-> `if`(n=0, 1, add(multinomial(n, n/d$d)*C(n, d), d=divisors(n))):
%p A231812 seq(a(n), n=0..25);
%t A231812 multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := If[n == 0, 1, Sum[multinomial[n, Array[n/d&, d]]*Binomial[n, d], {d, Divisors[n]}]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A231812 Main diagonal of A231915.
%Y A231812 Cf. A000312, A005095, A231807.
%K A231812 nonn
%O A231812 0,3
%A A231812 _Alois P. Heinz_, Nov 13 2013