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A245911 Number of pairs of endofunctions f, g on [n] satisfying f(g^n(i)) = f(i) for all i in [n].

%I A245911 #10 Oct 03 2019 11:19:27
%S A245911 1,1,12,207,9184,173225,46097856,729481375,454190410752,
%T A245911 30607186160529,12762075858688000,1036636706945881151,
%U A245911 3080713389889966460928,145084860487902521548921,124325137916420574135066624,56537825009822523196823829375
%N A245911 Number of pairs of endofunctions f, g on [n] satisfying f(g^n(i)) = f(i) for all i in [n].
%H A245911 Alois P. Heinz, <a href="/A245911/b245911.txt">Table of n, a(n) for n = 0..100</a>
%p A245911 with(combinat):
%p A245911 T:= proc(n, j) T(n, j):= binomial(n-1, j-1)*n^(n-j) end:
%p A245911 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, x^n,
%p A245911       expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A245911       x^(igcd(i, k)*j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A245911     end:
%p A245911 a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
%p A245911         i=0..degree(p)))(b(j$2, n)), j=0..n):
%p A245911 seq(a(n), n=0..20);
%t A245911 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A245911 b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i - 1)!^j*multinomial[n, Join[{n - i*j}, Array[i&, j]]]/j!* x^(GCD[i, k]*j)*b[n - i*j, i - 1, k][x], {j, 0, n/i}]]]];
%t A245911 a[n_] := If[n == 0, 1, Sum[Binomial[n - 1, j - 1]*n^(n - j)*b[j, j, n][n], {j, 0, n}]];
%t A245911 a /@ Range[0, 20] (* _Jean-François Alcover_, Oct 03 2019, after _Alois P. Heinz_ *)
%Y A245911 Main diagonal of A245910.
%Y A245911 Cf. A245988.
%K A245911 nonn
%O A245911 0,3
%A A245911 _Alois P. Heinz_, Aug 06 2014