A239761 Number of pairs of functions (f, g) on a set of n elements into itself satisfying f(g(x)) = f(x).

1, 1, 10, 159, 3496, 98345, 3373056, 136535455, 6371523712, 336784920849, 19888195110400, 1297716672601151, 92721494240225280, 7199830049013964921, 603715489091812335616, 54366622743565012989375, 5233114241479255004839936, 536180296483497244155041825

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..345

Formula

a(n) ~ 5^(-1/4) * ((1+sqrt(5))/2)^(3*n-1/2) * n^n / exp(2*n/(1+sqrt(5))). - Vaclav Kotesovec, Aug 07 2014

a(n) = Sum_{k = 1..n} A060281(n,k) n^k. - David Einstein, Oct 31 2016

a(n) = n! * [x^n] 1/(1 + LambertW(-x))^n. - Ilya Gutkovskiy, Oct 03 2017

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)*j^j, j=0..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 17 2014

Mathematica

f4[n_] := Sum[n^k Sum[Binomial[n - 1, j]*n^(n - 1 - j)*StirlingS1[j + 1, k] *(-1)^(j + k + 1), {j, 0, n - 1}], {k, 1, n}] (* David Einstein, Oct 31 2016 *)

Crossrefs

Cf. A181162, A239769, A239773.

Column k=1 of A245910.

Cf. A001622, A295188.

Sequence in context: A200989 A268883 A283721 * A245916 A245914 A239762

Adjacent sequences: A239758 A239759 A239760 * A239762 A239763 A239764

Keyword

nonn

Author

Chad Brewbaker, Mar 26 2014

Extensions

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

a(8)-a(17) from Alois P. Heinz, Jul 17 2014

Status

approved