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 1-5 of 5 results.

A372821 Table read by antidiagonals: T(m,n) = number of (m-2)-metered (m,n)-parking functions.

Original entry on oeis.org

0, 1, 0, 0, 4, 0, 0, 4, 9, 0, 0, 0, 21, 16, 0, 0, 0, 27, 56, 25, 0, 0, 0, 0, 163, 115, 36, 0, 0, 0, 0, 257, 483, 204, 49, 0, 0, 0, 0, 0, 1686, 1095, 329, 64, 0, 0, 0, 0, 0, 3156, 5367, 2131, 496, 81, 0, 0, 0, 0, 0, 0, 21858, 13076, 3747, 711, 100, 0, 0, 0, 0, 0, 0, 47442, 73276, 27309, 6123, 980, 121, 0
Offset: 1

Views

Author

Spencer Daugherty, May 13 2024

Keywords

Examples

			Table begins:
  0, 0, 0, 0, 0, 0, 0, ...
  1, 4, 9, 16, 25, 36, 49, ...
  0, 4, 21, 56, 115, 204, 329, ...
  0, 0, 27, 163, 483, 1095, 2131, ...
  0, 0, 0, 257, 1686, 5367, 13076, ...
  0, 0, 0, 0, 3156, 21858, 73276, ...
  0, 0, 0, 0, 0, 47442, 341192, ...
  ...

Links

  • Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Crossrefs

Main diagonal is A328694.
Cf. A372817, A372818, A372819, A372820.

Formula

T(m,n) = (n-m+2)^2*(m-1)^(m-3) + Sum_{k=n-m+3...n} binomial(m-2, n-k)*(n-k+1)^(n-k-1)*[binomial(k+1,2)*(n+m+2)*k^(m-n+k-3) + (k*(n-m+1) - binomial(n-m+2,2))*(k-n+m-1)^(k-n+m-3) + Sum_{j=n-m+2} (jk - binomial(j+1,2))*binomial(m-2-n+k, k-1-j)*(n-m+1)*j^(j+m-2-n)*(k-j)^(k-j-2)].

A372819 Table read by antidiagonals: T(m,n) = number of 3-metered (m,n)-parking functions.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 0, 8, 4, 0, 0, 16, 15, 5, 0, 0, 0, 50, 24, 6, 0, 0, 0, 125, 108, 35, 7, 0, 0, 0, 257, 432, 196, 48, 8, 0, 0, 0, 540, 1686, 1029, 320, 63, 9, 0, 0, 0, 1200, 6253, 5367, 2048, 486, 80, 10, 0, 0, 0, 3000, 23228, 27629, 13076, 3645, 700, 99, 11
Offset: 1

Views

Author

Spencer Daugherty, May 13 2024

Keywords

Examples

			Table beings:
  1, 2,  3,    4,     5,      6,      7, ...
  0, 3,  8,   15,    24,     35,     48, ...
  0, 0, 16,   50,   108,    196,    320, ...
  0, 0,  0,  125,   432,   1029,   2048, ...
  0, 0,  0,  257,  1686,   5367,  13076, ...
  0, 0,  0,  540,  6253,  27629,  83069, ...
  0, 0,  0, 1200, 23228, 140599, 525594, ...
  ...

Links

  • Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Crossrefs

Main diagonal is third row of A372816.
Cf. A372818, A372820, A372821, A005563.

A372816 Table read by antidiagonals: T(t,n) = number of t-metered parking functions of length n.

Original entry on oeis.org

1, 1, 3, 1, 3, 21, 1, 3, 16, 209, 1, 3, 16, 163, 2640, 1, 3, 16, 125, 2142, 40391, 1, 3, 16, 125, 1686, 33961, 726103, 1, 3, 16, 125, 1296, 27629, 626569, 15003009, 1, 3, 16, 125, 1296, 21858, 525594, 13198604, 350382231
Offset: 1

Views

Author

Spencer Daugherty, May 13 2024

Keywords

Examples

			Table begins:
  1, 3, 21, 209, 2640, 40391, 726103, ...
  1, 3, 16, 163, 2142, 33961, 626569, ...
  1, 3, 16, 125, 1686, 27629, 525594, ...
  1, 3, 16, 125, 1296, 21858, 430062, ...
  1, 3, 16, 125, 1296, 16807, 341192, ...
  1, 3, 16, 125, 1296, 16807, 262144, ...
  ...

Links

  • Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Crossrefs

First row is A097690 and main diagonal of A372817.
Second, third and fourth rows are main diagonals of A372818, A372819, and A372820.
Cf. A372821, A000272.

A372817 Table read by antidiagonals: T(m,n) = number of 1-metered (m,n)-parking functions.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 4, 8, 4, 0, 6, 21, 15, 5, 0, 8, 55, 56, 24, 6, 0, 12, 145, 209, 115, 35, 7, 0, 16, 380, 780, 551, 204, 48, 8, 0, 24, 1000, 2912, 2640, 1189, 329, 63, 9, 0, 32, 2625, 10868, 12649, 6930, 2255, 496, 80, 10, 0, 48, 6900, 40569, 60606, 40391, 15456, 3905, 711, 99, 11
Offset: 1

Views

Author

Spencer Daugherty, May 13 2024

Keywords

Examples

			For T(3,2) the 1-metered (3,2)-parking functions are 111, 121, 211, 212.
Table begins:
  1,  2,    3,     4,     5,      6,      7, ...
  0,  3,    8,    15,    24,     35,     48, ...
  0,  4,   21,    56,   115,    204,    329, ...
  0,  6,   55,   209,   551,   1189,   2255, ...
  0,  8,  145,   780,  2640,   6930,  15456, ...
  0, 12,  380,  2912, 12649,  40391, 105937, ...
  0, 16, 1000, 10868, 60606, 235416, 726103, ...
  ...

Links

  • Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Crossrefs

Main diagonal is A097690 and first row of A372816.
First, second, and third diagonals above main are A097691, A342167, A342168.
Second column A029744. Second row A005563. Third row A242135.
Cf. A001353, A004254, A001109, A004187, A372818, A372821.

Formula

T(m,n) = (n*(n+sqrt(n^2 - 4))-2)/(n*(n+sqrt(n^2 - 4))-4)*((n+sqrt(n^2-4))/2)^m + (n*(n-sqrt(n^2 - 4))-2)/(n*(n-sqrt(n^2 - 4))-4)*((n-sqrt(n^2-4))/2)^m.
T(m,n) = n*T(m-1,n) - T(m-2,n) with T(0,n) = 1.

A374738 Table read by ascending antidiagonals: T(m,n) = number of (n-1)-metered (m,n)-parking functions.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 6, 16, 15, 5, 1, 8, 27, 50, 24, 6, 1, 12, 48, 125, 108, 35, 7, 1, 16, 96, 257, 432, 196, 48, 8, 1, 24, 162, 540, 1296, 1029, 320, 63, 9, 1, 32, 288, 1200, 3156, 4802, 2048, 486, 80, 10, 1, 48, 576, 3000, 7734, 16807, 12288, 3645, 700, 99, 11
Offset: 1

Views

Author

Spencer Daugherty, Jul 18 2024

Keywords

Examples

			Table begins:
   1,  2,   3,    4,    5,     6,      7, ...
   1,  3,   8,   15,   24,    35,     48, ...
   1,  4,  16,   50,  108,   196,    320, ...
   1,  6,  27,  125,  432,  1029,   2048, ...
   1,  8,  48,  257, 1296,  4802,  12288, ...
   1, 12,  96,  540, 3156, 16807,  65536, ...
   1, 16, 162, 1200, 7734, 47442, 262144, ...
   ...

Links

  • Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Crossrefs

The n=m+1 diagonal is A007334.
Cf. A372817, A372818, A372819, A372820, A372821.

Formula

T(n+k,n) = Sum_{sigma = (sigma_1, ..., sigma_n) in S_n} (( Product_{i=1..n} L_{i}(sigma))( Product_{j=1..k} sigma_j mod n )), where k>0 and L_{i}(sigma) is the largest index h with i= sigma_N for all N in {i-j, i-j+1, ..., i-1, i}.
Showing 1-5 of 5 results.

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