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A231812 Number of endofunctions on [n] where all nonempty preimages have the same cardinality. 4
1, 1, 4, 9, 64, 125, 2826, 5047, 218688, 504009, 32216950, 39916811, 7585223196, 6227020813, 2424646536326, 1813027195995, 1072898135852416, 355687428096017, 616925243565037854, 121645100408832019, 441395941479128984940, 72313131901887676821 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of endofunctions f:{1,...,n}-> {1,...,n} such that (1<=i<j<=n and |f^(-1)(i)|>0 and |f^(-1)(j)|>0) implies |f^(-1)(i)| = |f^(-1)(j)|.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

FORMULA

a(n) = Sum_{d|n} multinomial(n; {n/d}^d)*C(n,d) for n>0, a(0) = 1.

a(n) = n! + n = A005095(n) for prime n.

EXAMPLE

a(2) = 4: (1,1), (1,2), (2,1), (2,2).

a(3) = 9: (1,1,1), (1,2,3), (1,3,2), (2,1,3), (2,2,2), (2,3,1), (3,1,2), (3,2,1), (3,3,3).

a(4) = 64: (1,1,1,1), (1,1,2,2), (1,1,3,3), ..., (4,4,3,3), (4,4,4,4).

MAPLE

with(numtheory): with(combinat): C:= binomial:

a:= n-> `if`(n=0, 1, add(multinomial(n, n/d$d)*C(n, d), d=divisors(n))):

seq(a(n), n=0..25);

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := If[n == 0, 1, Sum[multinomial[n, Array[n/d&, d]]*Binomial[n, d], {d, Divisors[n]}]]; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Dec 27 2013, translated from Maple *)

CROSSREFS

Main diagonal of A231915.

Cf. A000312, A005095, A231807.

Sequence in context: A140483 A124683 A077163 * A069711 A062067 A110256

Adjacent sequences:  A231809 A231810 A231811 * A231813 A231814 A231815

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 13 2013

STATUS

approved

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Last modified October 14 18:28 EDT 2019. Contains 328022 sequences. (Running on oeis4.)