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

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

1,5

Links

Table of n, a(n) for n=1..78.

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

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)].

Example

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, ...
  ...

Crossrefs

Main diagonal is A328694.

Cf. A372817, A372818, A372819, A372820.

Sequence in context: A308256 A096406 A189885 * A151673 A005397 A259124

Adjacent sequences: A372818 A372819 A372820 * A372822 A372823 A372824

Keyword

nonn,tabl

Author

Spencer Daugherty, May 13 2024

Status

approved