|
|
A208250
|
|
The sum of the largest preimage over all functions f:{1,2,...,n}->{1,2,...,n}.
|
|
2
|
|
|
0, 1, 6, 51, 544, 7145, 112356, 2066323, 43574336, 1036922769, 27486891100, 803137535321, 25642631336400, 888148407804853, 33165208812574216, 1328185604750416875, 56783630865774075136, 2581268127178259819297, 124322489582200453748268, 6324172127062894070727625
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
n labeled balls are placed in n labeled urns. The maximum number of balls in an urn is summed over all n^n possible configurations. a(n) is this sum.
|
|
REFERENCES
|
R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison and Wesley, 1996, page 435.
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: Sum_{j>=0} exp(x)^n - (Sum_{i=0..j} x^i/i!)^n.
a(n) ~ n^n log n/log log n. More precisely, a(n)/n^n = Gamma^(-1)(n) - 3/2 + o(1) where Gamma^(-1) is the inverse of the gamma function. See Gönnet section 4 or Mitzenmacher et al. - Charles R Greathouse IV, Feb 20 2013
|
|
EXAMPLE
|
a(2) = 6. The functions f:{1,2}->{1,2} written as words are: 11, 12, 21, 22 and we sum respectively 2 + 1 + 1 + 2 = 6.
|
|
MATHEMATICA
|
f[n_] := n! Coefficient[ Series[ Sum[ Exp[n*x] - Sum[x^i/i!, {i, 0, j}]^n, {j, 0, n}], {x, 0, n}], x^n]; f[0] = 0; Array[f, 19, 0] (* modified by Robert G. Wilson v, Feb 20 2013 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|