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

Author

Stefano Spezia, Nov 02 2019

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

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.

Crossrefs

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

Programs

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}]]

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).