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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
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 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.

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))/(1-T(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

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Last modified December 7 08:17 EST 2019. Contains 329841 sequences. (Running on oeis4.)