A055865 -id:A055865 - 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).

A055864 Coefficient triangle for certain polynomials.

Original entry on oeis.org

1, 3, 2, 16, 12, 9, 125, 100, 80, 64, 1296, 1080, 900, 750, 625, 16807, 14406, 12348, 10584, 9072, 7776, 262144, 229376, 200704, 175616, 153664, 134456, 117649, 4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152

Offset: 1

Wolfdieter Lang, Jun 20 2000

The coefficients of the partner polynomials are found in triangle A055858.

			Fourth row polynomial (n=4): p(4,x) = 125+100*x+80*x^2+64*x^3.
Triangle begins:
        1;
        3,       2;
       16,      12,       9;
      125,     100,      80,      64;
     1296,    1080,     900,     750,     625;
    16807,   14406,   12348,   10584,    9072,    7776;
   262144,  229376,  200704,  175616,  153664,  134456,  117649;
  4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152;
  ...

Column sequences are: A000272(n+1), n >= 1, A055865, A055070, A055867, A055868 for m=1..5.

Main diagonal gives A000169.

a[n_, m_] /; n < m = 0; a[n_, m_] := n^(m-1)*(n+1)^(n-m); Table[a[n, m], {n, 1, 8}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2013 *)

a(n, m)=0 if n= m >= 1;

E.g.f. for column m: A(m, x); A(1, x)=-(W(-x)/x+1); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-(((m-1)^(m-1))/m)* (x^(m-1))/(m-1)!, m >= 2; W(x) principal branch of Lambert's function.

A085389 a(1) = 1; for n >= 2, a(n) = (n*(n+1)^(n-1))/(n+1).

Original entry on oeis.org

1, 2, 12, 100, 1080, 14406, 229376, 4251528, 90000000, 2143588810, 56757583872, 1654301902188, 52644347205632, 1816448730468750, 67553994410557440, 2694045224950414864, 114692890480116793344, 5191945444217181018258, 249036800000000000000000, 12617615847934310595791220

Offset: 1

Paul Barry, Jun 30 2003

Paolo Xausa, Table of n, a(n) for n = 1..350
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A053506, A085390.

A085389[n_] := If[n == 1, 1, n*(n+1)^(n-2)];
Array[A085389, 25] (* Paolo Xausa, Aug 07 2025 *)

Main subdiagonal of A085388.
a(n) = A055865(n), n>1. - R. J. Mathar, Sep 12 2008
a(n) = [x^n] x*(1 - x)/(1 - x - n*x). - Ilya Gutkovskiy, Oct 02 2017

Name edited by Paolo Xausa, Aug 07 2025

A055070 Third column of triangle A055864.

Original entry on oeis.org

0, 0, 9, 80, 900, 12348, 200704, 3779136, 81000000, 1948717100, 52027785216, 1527047909712, 48884036690944, 1695352148437500, 63331869759897600, 2535571976423919872, 108321063231221415936, 4918685157679434648876

Offset: 1

Wolfdieter Lang Jun 20 2000

Cf. A055864, A000272, A055865.

a(1)=0= a(2); a(n)= n^2*(n+1)^(n-3), n >= 3.
E.g.f. (3*W(-x)^2-2*W(-x)^3-3*x^2-8*x^3)/(12*x), W(x) principal branch of Lambert's function.
a(n)=A055864(n, 3).

A055867 Fourth column of triangle A055864.

Original entry on oeis.org

0, 0, 0, 64, 750, 10584, 175616, 3359232, 72900000, 1771561000, 47692136448, 1409582685888, 45392319784448, 1582328671875000, 59373627899904000, 2386420683693101056, 102303226385042448384, 4659806991485780193672

Offset: 1

Wolfdieter Lang, Jun 20 2000

a(n)=A055864(n, 4). Cf. A000272, A055865, A055070, A055866, A055858.

A055867:=n->n^3*(n+1)^(n-4): 0,0,0,seq(A055867(n), n=4..30); # Wesley Ivan Hurt, Jan 27 2017

a(i)=0 for i=1, 2, 3; a(n) = n^3*(n+1)^(n-4), n >= 4.

E.g.f.: (9*W(-x)^2-14*W(-x)^3+3*W(-x)^4-9*x^2-32*x^3-81*x^4)/(72*x), W(x) principal branch of Lambert's function.

A055868 Fifth column of triangle A055864.

Original entry on oeis.org

0, 0, 0, 0, 625, 9072, 153664, 2985984, 65610000, 1610510000, 43717791744, 1301153248512, 42150011228416, 1476840093750000, 55662776156160000, 2246042996417036288, 96619713808095645696, 4414553991933897025584

Offset: 1

Wolfdieter Lang Jun 20 2000

a(n)=A055864(n, 5). Cf. A000272, A055865, A055070, A055867, A055858.

a(i)=0 for i=1..4; a(n)= n^4*(n+1)^(n-5), n >= 5.

E.g.f.: (270*W(-x)^2-740*W(-x)^3+345*W(-x)^4-36*W(-x)^5-270*x^2-1280*x^3-3645*x^4-9216*x^5)/(4320*x), W(x) principal branch of Lambert's function.

A386011 Total number of inversions in all parking functions of length n.

Original entry on oeis.org

0, 1, 18, 300, 5400, 108045, 2408448, 59521392, 1620000000, 48230748225, 1560833556480, 54591962772204, 2053129541019648, 82648417236328125, 3546584706554265600, 161642713497024891840, 7799116552647941947392, 397183826482614347896737

Offset: 1

Kyle Celano, Jul 14 2025

			a(2)=1 because in the 3 parking functions of length 2 (11, 12, 21), there is 1 inversion: (1,2).

Kyle Celano, Table of n, a(n) for n = 1..100
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.
Richard P. Stanley, Parking Functions, 2011.
Wikipedia, Parking function.

Cf. A001809, A000272, A240796.

Table[Binomial[n,2] * n*(n+1)^(n-2)/2, {n, 0, 18}]

a(n) = binomial(n,2) * n*(n+1)^(n-2)/2.
a(n) = Sum_{k=0..binomial(n,2)} A152290(n,k)*k.
a(n) = binomial(n,2)*A055865(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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A055864 Coefficient triangle for certain polynomials.

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A085389 a(1) = 1; for n >= 2, a(n) = (n*(n+1)^(n-1))/(n+1).

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A055070 Third column of triangle A055864.

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A055867 Fourth column of triangle A055864.

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A055868 Fifth column of triangle A055864.

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A386011 Total number of inversions in all parking functions of length n.

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