A243594 Triangle read by rows: T(n,k) = coefficient of [x^(n-k)] in the expansion of the polynomial (x+n)^n.

1, 1, 1, 1, 4, 4, 1, 9, 27, 27, 1, 16, 96, 256, 256, 1, 25, 250, 1250, 3125, 3125, 1, 36, 540, 4320, 19440, 46656, 46656, 1, 49, 1029, 12005, 84035, 352947, 823543, 823543, 1, 64, 1792, 28672, 286720, 1835008, 7340032, 16777216, 16777216, 1, 81, 2916, 61236, 826686, 7440174, 44641044

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.

Links

Table of n, a(n) for n=0..51.

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

Cf. A071207 (row reversed).

Row sums give A000169.

Sequence in context: A080721 A123588 A289710 * A360707 A365674 A106026

Adjacent sequences: A243591 A243592 A243593 * A243595 A243596 A243597

Keyword

nonn,tabl

Author

Annette Kim, Jun 07 2014

Status

approved