A085389 -id:A085389 - cp's OEIS frontend

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9

Offset: 0

Alois P. Heinz, Aug 19 2013

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);

f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n}   T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A085388 First differences of n^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 8, 0, 1, 5, 20, 48, 54, 16, 0, 1, 6, 30, 100, 192, 162, 32, 0, 1, 7, 42, 180, 500, 768, 486, 64, 0, 1, 8, 56, 294, 1080, 2500, 3072, 1458, 128, 0, 1, 9, 72, 448, 2058, 6480, 12500, 12288, 4374, 256, 0, 1, 10, 90, 648

Offset: 1

Paul Barry, Jun 30 2003

T(n,k) is the number of k-digit numbers in base n; n,k >= 2. - Mohammed Yaseen, Nov 11 2022

			Rows begin
  1,   0,   0,   0,   0, ...
  1,   1,   2,   4,   8, ...
  1,   2,   6,  18,  54, ...
  1,   3,  12,  48, 192, ...
  1,   4,  20, 100, 500, ...

Rows include A011782, A025192, A002001, A005054, A052934, A055272, A055274, A055275, A052268, A055276.
Diagonals include A053506, A085389, A085390.
Row-wise binomial transform is A083064.

T(n,k) = (n-1)*n^(k-1) + 0^k/n. - Corrected by Mohammed Yaseen, Nov 11 2022

T(n,0) = 1; T(n,k) = n^k - n^(k-1) for k >= 1. - Mohammed Yaseen, Nov 11 2022

Offset corrected by Mohammed Yaseen, Nov 11 2022

A085390 a(n) = (n(n+1)^(n-2)+0^(n-2))/(n+1).

Original entry on oeis.org

1, 3, 20, 180, 2058, 28672, 472392, 9000000, 194871710, 4729798656, 127253992476, 3760310514688, 121096582031250, 4222124650659840, 158473248526494992, 6371827248895377408, 273260286537746369382, 12451840000000000000000

Offset: 2

Paul Barry, Jun 30 2003

A diagonal of square array A085388.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A085389, A053506.

A386015 Number of parking functions of size n with a descent in the first position.

Original entry on oeis.org

0, 1, 6, 50, 540, 7203, 114688, 2125764, 45000000, 1071794405, 28378791936, 827150951094, 26322173602816, 908224365234375, 33776997205278720, 1347022612475207432, 57346445240058396672, 2595972722108590509129, 124518400000000000000000, 6308807923967155297895610, 336682260736692839281065984

Offset: 1

Gabe Udell, Jul 14 2025

			For n=1 the only parking function is 1 and it does not have a descent in the first position so a(1)=0.
For n=2 there are 3 parking functions: 11,12,21. Among them only 21 has a descent in the first position so a(2)=1.
For n=3 there are 16 parking functions: 111,112,121,211,122,212,221,113,131,311,123,132,213,231,312,321. Of these, 211,212,311,213,312, and 321 have a descent in the first position so a(3)=6.

Paolo Xausa, Table of n, a(n) for n = 1..350
Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, Inversions in parking functions, arXiv:2508.11587 [math.CO], 2025.
Wikipedia, Parking function.

Cf. A000272, A085389.

A386015[n_] := If[n == 1, 0, n*(n+1)^(n-2)/2];
Array[A386015, 25] (* Paolo Xausa, Aug 07 2025 *)

a(n) = (n/2)*(n+1)^(n-2) for n >= 2.

a(n) = A085389(n) / 2 for n >= 2.

cp's OEIS Frontend

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A085388 First differences of n^k.

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A085390 a(n) = (n(n+1)^(n-2)+0^(n-2))/(n+1).

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A386015 Number of parking functions of size n with a descent in the first position.

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