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A239772 Number of pairs of functions f, g from a size n set into itself satisfying f(f(x)) = f(g(f(x))).

%I A239772 #16 Jan 10 2025 18:33:10
%S A239772 1,1,10,231,9688,603445,52284816,5951141035,856275088768,
%T A239772 151330313546361,32121886627244800,8043522214887251191,
%U A239772 2341436450503523834880,782684599861773582454741,297337340445195054893615104,127232791559907423447708979875,60852096942278280426353043275776,32309821732254010064727052008198385
%N A239772 Number of pairs of functions f, g from a size n set into itself satisfying f(f(x)) = f(g(f(x))).
%H A239772 Max Alekseyev, <a href="/A239772/b239772.txt">Table of n, a(n) for n = 0..100</a>
%F A239772 Formula is given in the Sage code. - _Max Alekseyev_, Jan 10 2025
%p A239772 s:= proc(n, i) option remember; `if`(i=0, [[]],
%p A239772        map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
%p A239772     end:
%p A239772 a:= proc(n) local l; l:= s(n$2);
%p A239772        add(add(`if`([seq(evalb(f[f[i]]=f[g[f[i]]]),
%p A239772        i=1..n)]=[true$n], 1, 0), g=l), f=l)
%p A239772     end:
%p A239772 seq(a(n), n=0..5);  # _Alois P. Heinz_, Jul 16 2014
%o A239772 (Sage)
%o A239772 def a239772(n):
%o A239772     L.<z> = LaurentPolynomialRing(QQ)
%o A239772     R.<y> = PowerSeriesRing(L, default_prec=n+1)
%o A239772     h = 1 - sum((y*(1+i*z))^i*n^(i-1)/factorial(i) for i in (1..n))//z
%o A239772     return h.inverse()[n][0] * factorial(n) # _Max Alekseyev_, Jan 10 2025
%Y A239772 Cf. A181162, A239769, A239773.
%K A239772 nonn
%O A239772 0,3
%A A239772 _Chad Brewbaker_, Mar 26 2014
%E A239772 a(6)-a(7) from _Giovanni Resta_, Mar 28 2014
%E A239772 Terms a(8) onward from _Max Alekseyev_, Jan 10 2025