OFFSET
0,5
COMMENTS
T(n,k) is the number of partial functions f on an n-set such that |dom(f)| = k. - Geoffrey Critzer, Jan 09 2023
REFERENCES
O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009.
FORMULA
T(n,k) = binomial(n,k)*n^k.
E.g.f.: 1/(1-t(y*x))*exp(x*exp(t(y*x))) where t(z) is the e.g.f. for A000169. - Geoffrey Critzer, Jan 09 2023
EXAMPLE
n = 0, x^ 0 = 1;
n = 1, (x+1)^1 = x + 1, so 1, 1;
n = 2, (x+2)^2 = x^2 + 4x + 4, so 1, 4, 4;
n = 3, (x+3)^3 = x^3 + 9x^2 + 27x + 27, so 1, 9, 27, 27.
Triangle starts:
1;
1, 1;
1, 4, 4;
1, 9, 27, 27;
1, 16, 96, 256, 256;
1, 25, 250, 1250, 3125, 3125;
...
MAPLE
A243594 := proc(n, k)
expand( (x+n)^n) ;
coeftayl(%, x=0, n-k) ;
end proc:
for n from 0 to 12 do
for k from 0 to n do
printf("%d, ", A243594(n, k)) ;
end do:
end do: # R. J. Mathar, Jun 10 2014
MATHEMATICA
nn = 6; t[x_] := Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Map[Select[#, # > 0 &] &,
Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t[ y x])]] Exp[x Exp[ t[y x]]], {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jan 09 2023 *)
PROG
(PARI) row(n) = {my(p = (1+n*'x)^n); for (k=0, poldegree(p), print1(polcoeff(p, k), ", ")); } \\ Michel Marcus, Jun 10 2014
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Annette Kim, Jun 07 2014
STATUS
approved