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

1, 1, 10, 231, 9688, 603445, 52284816, 5951141035, 856275088768, 151330313546361, 32121886627244800, 8043522214887251191, 2341436450503523834880, 782684599861773582454741, 297337340445195054893615104, 127232791559907423447708979875, 60852096942278280426353043275776, 32309821732254010064727052008198385

Offset: 0

Author

Chad Brewbaker, Mar 26 2014

Keywords

• nonn

Links

• Max Alekseyev, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A181162, A239769, A239773.

Programs

Maple

s:= proc(n, i) option remember; `if`(i=0, [[]],
       map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
    end:
a:= proc(n) local l; l:= s(n$2);
       add(add(`if`([seq(evalb(f[f[i]]=f[g[f[i]]]),
       i=1..n)]=[true$n], 1, 0), g=l), f=l)
    end:
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 16 2014

Sage

def a239772(n):
    L. = LaurentPolynomialRing(QQ)
    R. = PowerSeriesRing(L, default_prec=n+1)
    h = 1 - sum((y*(1+i*z))^i*n^(i-1)/factorial(i) for i in (1..n))//z
    return h.inverse()[n][0] * factorial(n) # Max Alekseyev, Jan 10 2025

Formula

Formula is given in the Sage code. - Max Alekseyev, Jan 10 2025

Extensions

a(6)-a(7) from Giovanni Resta, Mar 28 2014

Terms a(8) onward from Max Alekseyev, Jan 10 2025