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 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

%I

%S 1,1,1,3,2,4,16,9,12,27,125,64,72,108,256,1296,625,640,810,1280,3125,

%T 16807,7776,7500,8640,11520,18750,46656,262144,117649,108864,118125,

%U 143360,196875,326592,823543,4782969,2097152,1882384,1959552,2240000,2800000,3919104,6588344,16777216

%N 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.

%C Here, for any x in the domain of definition (f^i)(x) denotes the i-fold 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.

%C T(n,n) = n^n, the partial functions that are total functions.

%C T(n,0) = A000272(offset), see comment and link by _Dennis P. Walsh_.

%H G. C. Greubel, <a href="/A185390/b185390.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 132, II.21.

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

%e Triangle begins

%e 1;

%e 1, 1;

%e 3, 2, 4;

%e 16, 9, 12, 27;

%e 125, 64, 72, 108, 256;

%e 1296, 625, 640, 810, 1280, 3125;

%e 16807, 7776, 7500, 8640, 11520, 18750, 46656;

%t 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

%Y Cf. A000169, A000272.

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Feb 09 2012

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Last modified April 19 08:43 EDT 2019. Contains 322241 sequences. (Running on oeis4.)