A239752 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = g(f(f(x))).

1, 1, 6, 123, 4792, 294645, 26351856, 3213829339, 511432765824, 102813166760265, 25450790212460800, 7599894406225438911, 2691706949197641133056, 1114869818722491048119773, 533573397145124307890731008, 292063395009538745067415219875, 181221082305680372426427865071616

Offset

0,3

Comments

Also pairs of functions where f(f(f(x))) = g(f(f(x))).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..160

Formula

a(n) = n! Sum_{d = 0..n} n^(n-d) Sum_{c = 0..(n-d)/2} S2(c+d,d) Sum_{b = c..n-c-d} S2(b,c)/(b! (n-b-c-d)!) d^(n-b-c-d). - David Einstein, Oct 23 2016

Mathematica

t1[n_, d_, c_] := StirlingS2[c + d, d] Sum[ StirlingS2[b, c]/(b! (n - b - c - d)!) d^(n - b - c - d), {b, c, n - c - d}]
a[n_] := If[n == 0, 1, n! Sum[n^(n - d) Sum[t1[n, d, c], {c, 0, Floor[(n - d)/2]}], {d, 1, n}]]
Table[a[n], {n, 0, 20}] (* David Einstein, Oct 23 2016 *)

Crossrefs

Cf. A000248, A000949.

Sequence in context: A138572 A356199 A078261 * A193223 A372920 A132507

Adjacent sequences: A239749 A239750 A239751 * A239753 A239754 A239755

Keyword

nonn

Author

Chad Brewbaker, Mar 26 2014

Extensions

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

a(8)-a(10) from Paul Boddington, Feb 23 2015

a(11)-a(16) from David Einstein, Oct 23 2016

Status

approved