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

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

Author

Spencer Daugherty, May 13 2024

Keywords

• nonn
• tabl

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.