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
Geoffrey Critzer, Distribution of non-functional points under a random partial function
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.
T(n,k) = binomial(n,k)*k^k*(n-k+1)^(n-k-1). - Geoffrey Critzer, Feb 28 2022
Sum_{k=0..n} k * T(n,k) = A185391(n). - Alois P. Heinz, Jan 12 2024
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;
...
MAPLE
T:= (n, k)-> binomial(n, k)*k^k*(n-k+1)^(n-k-1):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Jan 12 2024
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
PROG
(Julia)
T(n, k) = binomial(n, k)*k^k*(n-k+1)^(n-k-1)
for n in 0:9 (println([T(n, k) for k in 0:n])) end
# Peter Luschny, Jan 12 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 09 2012
STATUS
approved