

A185390


Triangular array read by rows. T(n,k) is the number of partial functions on n labeled objects in which the domain of definition contains exactly k elements such that for all i in {1,2,3,...}, (f^i)(x) is defined.


3



1, 1, 1, 3, 2, 4, 16, 9, 12, 27, 125, 64, 72, 108, 256, 1296, 625, 640, 810, 1280, 3125, 16807, 7776, 7500, 8640, 11520, 18750, 46656, 262144, 117649, 108864, 118125, 143360, 196875, 326592, 823543, 4782969, 2097152, 1882384, 1959552, 2240000, 2800000, 3919104, 6588344, 16777216
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

Here, for any x in the domain of definition (f^i)(x) denotes the ifold composition of f with itself, e.g., (f^2)(x) = f(f(x)). The domain of definition is the set of all values x for which f(x) is defined.
T(n,n) = n^n, the partial functions that are total functions.
T(n,0) = A000272(offset), see comment and link by Dennis P. Walsh.


LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132, II.21.


FORMULA

E.g.f.: exp(T(x))/(1T(x*y)) where T(x) is the e.g.f. for A000169.


EXAMPLE

Triangle begins
1;
1, 1;
3, 2, 4;
16, 9, 12, 27;
125, 64, 72, 108, 256;
1296, 625, 640, 810, 1280, 3125;
16807, 7776, 7500, 8640, 11520, 18750, 46656;


MATHEMATICA

nn = 7; tx = Sum[n^(n  1) x^n/n!, {n, 1, nn}]; txy = Sum[n^(n  1) (x y)^n/n!, {n, 1, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, Range[0, nn]! CoefficientList[Series[Exp[tx]/(1  txy), {x, 0, nn}], {x, y}]] // Flatten


CROSSREFS

Cf. A000169, A000272.
Sequence in context: A137824 A019321 A279261 * A019116 A213611 A318304
Adjacent sequences: A185387 A185388 A185389 * A185391 A185392 A185393


KEYWORD

nonn,tabl


AUTHOR

Geoffrey Critzer, Feb 09 2012


STATUS

approved



