A174551 -id:A174551 - cp's OEIS frontend

A174552 Triangular array T(n,k): The differences in the columns of A174551.

1, 0, 1, 1, 1, 2, 8, 7, 6, 6, 81, 65, 50, 36, 24, 1024, 781, 570, 390, 240, 120, 15625, 11529, 8162, 5460, 3360, 1800, 720, 279936, 201811, 140070, 92526, 57120, 31920, 15120, 5040

Offset: 0

Geoffrey Critzer, Mar 22 2010

T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the image of f contains {1,2,...,k} but not k+1.
Row sums = n^n.
Columns are asymptotic to n^n (1/e)(1-(1/e))^k.
Sum k*T(n,k) appears to be A055869.

			Triangle begins:
1;
0, 1;
1, 1, 2;
8, 7, 6, 6;
81,65,50,36,24;
1024,781,570,390,240,120;
15625, 11529, 8162, 5460, 3360, 1800, 720...

Table[Append[(-1) Differences[ Table[Sum[(-1)^i Binomial[k, i] (n - i)^n, {i, 0, k}], {k, 0, n}]], n! ], {n, 0, 7}] // Grid

A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 4, 6, 6, 27, 38, 36, 24, 256, 350, 330, 240, 120, 3125, 4202, 3960, 3000, 1800, 720, 46656, 62062, 58506, 45360, 29400, 15120, 5040, 823543, 1087214, 1025388, 806904, 546000, 312480, 141120, 40320, 16777216, 22024830, 20781690, 16524144, 11493720, 6985440, 3598560, 1451520, 362880

Offset: 1

Rui Duarte, Jan 22 2017

A parking function of size n is a sequence (a_1,...,a_n) of positive integers such that, if b_1 <= b_2 <= ... <= b_n is the increasing rearrangement of the sequence (a_1,..,a_n), then b_i <= i.
Given a:[n]->[n], the center of a is the largest subset Z(a) = { z_1, ..., z_k } of [n] such that z_1 < z_2 < ... < z_k and a_(z_j) <= j, for every j in [k]. The length of the center of a is |Z(a)|.
Then T(n,k)= number of parking functions of size n with center of length k.

			First seven rows:
      1
      1      2
      4      6      6
     27     38     36     24
    256    350    330    240    120
   3125   4202   3960   3000   1800    720
  46656  62062  58506  45360  29400  15120   5040

Rui Duarte and António Guedes de Oliveira, The number of parking functions with center of a given length, arXiv:1611.03707 (2016).

T(n,k) = k * A174551(n-1,k-1).
T(n,1) = (n-1)^(n-1) = A000312(n-1).
T(n,n-1) = n!(n-1)/2 = A001286(n), n>=2.
T(n,n) = n! = A000142(n).
Sum_{i=1,...,n} T(n,i) = (n+1)^(n-1) = A000272(n+1).

Table[Which[n == k == 1, 1, k == 1, (n - 1)^(n - 1), k == n, n!, True, k Sum[(-1)^j*Binomial[k - 1, j] (n - 1 - j)^(n - 1), {j, 0, k - 1}]], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 23 2017 *)

T(n,k) = k*Sum_{j=0..k-1} (-1)^j*binomial(k-1,j)*(n-1-j)^(n-1).

T(n,k) = k!*Sum_{j_1+j_2+...+j_k=n-k} (n-1)^(j_1)*(n-2)^(j_2)*...*(n-k)^(j_k).

cp's OEIS Frontend

A174552 Triangular array T(n,k): The differences in the columns of A174551.

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