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

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
4  | 14  35  75 125 125

Mathematica

T[r_, i_, k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r, i, 1], {r, 0, 9}, {i, 0, r}]]

Crossrefs

Cf. A000108, A000272, A007318, A329058, A329059, A329060, A329096 (row sums).

Sequence in context: A296674 A297073 A019460 * A236165 A049855 A286868

Adjacent sequences: A329054 A329055 A329056 * A329058 A329059 A329060

Keyword

nonn,tabl

Author

Stefano Spezia, Nov 02 2019

Status

approved