A329057 - OEIS

A329057

1-parking triangle T(r, i, 1) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 1 and 0 <= i <= r.

1, 1, 1, 2, 3, 3, 5, 10, 16, 16, 14, 35, 75, 125, 125, 42, 126, 336, 756, 1296, 1296, 132, 462, 1470, 4116, 9604, 16807, 16807, 429, 1716, 6336, 21120, 61440, 147456, 262144, 262144, 1430, 6435, 27027, 104247, 360855, 1082565, 2657205, 4782969, 4782969, 4862, 24310, 114400, 500500, 2002000, 7150000, 22000000, 55000000, 100000000, 100000000

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The k-parking numbers interpolate between the generalized Fuss-Catalan numbers and the number of parking functions (see Yip).

LINKS

Stefano Spezia, First 151 rows of the triangle, flattened

Carolina Benedetti, Rafael S. González D’León, Christopher R. H. Hanusa, Pamela E. Harris, Apoorva Khare, Alejandro H. Morales, Martha Yip, The volume of the caracol polytope, Séminaire Lotharingien de Combinatoire 80B.87 (2018).

Martha Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

FORMULA

T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i).

T(r, 0, 1) = A000108(r).

T(r, r, 1) = A000272(r + 1).

EXAMPLE

r/i| 0 1 2 3 4

———————————————————————

0 | 1

1 | 1 1

2 | 2 3 3

3 | 5 10 16 16