cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.

Original entry on oeis.org

1, 4, 2, 24, 15, 9, 200, 136, 100, 64, 2160, 1535, 1215, 945, 625, 28812, 21036, 17286, 14406, 11526, 7776, 458752, 341103, 286671, 247296, 211456, 172081, 117649, 8503056, 6405904, 5464712, 4811528, 4251528, 3691528, 3038344, 2097152, 180000000, 136953279, 118078911, 105372819, 94921875, 85078125, 74627181, 61921089, 43046721
Offset: 1

Views

Author

Rui Duarte, Jan 22 2018

Keywords

Comments

T(n,k) is the number of pairs (f,i) such that f is a parking function and f(i) = k.

Examples

			Triangle begins:
====================================================================
n\k|       1       2       3       4       5       6       7       8
---|----------------------------------------------------------------
1  |       1
2  |       4       2
3  |      24      15       9
4  |     200     136     100      64
5  |    2160    1535    1215     945     625
6  |   28812   21036   17286   14406   11526    7776
7  |  458752  341103  286671  247296  211456  172081  117649
8  | 8503056 6405904 5464712 4811528 4251528 3691528 3038344 2097152
  ...

Crossrefs

Cf. A000169, A000272, A089946, A298592, A298594, A298597.

Programs

  • Mathematica
    Table[n Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

Formula

T(n,k) = n*Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).
T(n,k) = n*A298592(n,k).
T(n,k) = n*Sum_{j=k..n} A298594(n,j).
T(n,k) = Sum_{j=k..n} A298597(n,j).
Sum_{k=1..n} T(n,k) = n*A000272(n+1).
T(n+1,1) = A089946(n), T(n,n) = A000169(n). - Andrey Zabolotskiy, Feb 21 2018

A298594 Triangle read by rows: T(n,k) = number of parking functions a of length n such that a(1) = k and if we replace a(1) = k with k+1 we don't get a parking function.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 16, 9, 9, 16, 125, 64, 54, 64, 125, 1296, 625, 480, 480, 625, 1296, 16807, 7776, 5625, 5120, 5625, 7776, 16807, 262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144, 4782969, 2097152, 1411788, 1161216, 1093750, 1161216, 1411788, 2097152, 4782969
Offset: 1

Views

Author

Rui Duarte, Jan 22 2018

Keywords

Examples

			Triangle begins:
       1;
       1,      1;
       3,      2,     3;
      16,      9,     9,    16;
     125,     64,    54,    64,   125;
    1296,    625,   480,   480,   625,  1296;
   16807,   7776,  5625,  5120,  5625,  7776,  16807;
  262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144;
  ...

Links

Crossrefs

Cf. A000272, A298592, A298593, A298597.

Programs

  • Mathematica
    Table[Binomial[n - 1, k - 1] k^(k - 2)*(n + 1 - k)^(n - 1 - k), {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

Formula

T(n,k) = binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).
T(n,k) = A298592(n,k) - A298592(n,k+1).
T(n,k) = (A298593(n,k) - A298593(n,k+1))/n.
T(n,k) = A298597(n,k)/n.
T(n,1) = A000272(n+2).
T(n,n) = A000272(n+2).
T(n,k) = T(n,n-k).

A298597 Number T(n,k) of times the value k appears on the parking functions of length n and such that if we replace that value k with k+1 we don't get a parking function.

Original entry on oeis.org

1, 2, 2, 9, 6, 9, 64, 36, 36, 64, 625, 320, 270, 320, 625, 7776, 3750, 2880, 2880, 3750, 7776, 117649, 54432, 39375, 35840, 39375, 54432, 117649, 2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152, 43046721, 18874368, 12706092, 10450944, 9843750, 10450944, 12706092, 18874368, 43046721
Offset: 1

Views

Author

Rui Duarte, Jan 22 2018

Keywords

Examples

			Triangle begins:
        1;
        2,      2;
        9,      6,      9;
       64,     36,     36,     64;
      625,    320,    270,    320,    625;
     7776,   3750,   2880,   2880,   3750,   7776;
   117649,  54432,  39375,  35840,  39375,  54432, 117649;
  2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152;
  ...

Crossrefs

Cf. A000169, A000272, A298592, A298593, A298594.

Programs

  • Mathematica
    Table[n Binomial[n - 1, k - 1] k^(k - 2)*(n + 1 - k)^(n - 1 - k), {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

Formula

T(n,k) = n*binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).
T(n,k) = n*A298594(n,k).
T(n.k) = A298593(n,k)-A298593(n,k+1).
T(n,k) = n*(A298592(n,k)-A298592(n,k+1)).
T(n,1) = n*A000272(n+2).
T(n,n) = n*A000272(n+2).
T(n,1) = A000169(n).
T(n,n) = A000169(n).
T(n,k) = T(n,n-k).

A328694 a(n) = sum of lead terms of all parking functions of length n.

Original entry on oeis.org

1, 4, 27, 257, 3156, 47442, 843352, 17300943, 402210240, 10448526896, 299925224064, 9426724628301, 321959469056512, 11872685912032350, 470132249600142336, 19895288956008203963, 896055382220853362688, 42793946679993786078108, 2160123874888094765056000
Offset: 1

Views

Author

Andrew Howroyd, Oct 25 2019

Keywords

Examples

			Case n = 2: There are 3 parking functions of length 2: [1, 1], [1, 2], [2, 1]. Summing up the initial values gives 1 + 1 + 2 = 4, so a(2) = 4.

Links

Crossrefs

Cf. A298592, A318047.

Programs

  • PARI
    \\ here T(n,k) is A298592(n,k).
T(n, k)={sum(j=k, n, binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j))}
a(n)={sum(k=1, n, k*T(n, k))}

Formula

a(n) = Sum_{k=1..n} k*A298592(n,k).
a(n) = A318047(n) / n.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.