A329059 3-parking triangle T(r, i, 3) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 3 and 0 <= i <= r.

1, 3, 1, 15, 9, 3, 91, 78, 48, 16, 612, 680, 600, 375, 125, 4389, 5985, 6840, 6156, 3888, 1296, 32890, 53130, 74382, 86779, 79233, 50421, 16807, 254475, 475020, 786240, 1123200, 1331200, 1228800, 786432, 262144, 2017356, 4272048, 8155728, 13762791, 19978245, 23973894, 22320522, 14348907, 4782969

Offset: 0

Author

Stefano Spezia, Nov 03 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  |      3      1
2  |     15      9      3
3  |     91     78     48     16
4  |    612    680    600    375    125
...
		

Links

• Stefano Spezia, First 151 rows of the triangle, flattened
• Martha Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Crossrefs

Cf. A000108, A000272, A006632, A007318, A328978 (row sums), A329057, A329058, A329060.

Programs

Mathematica

T[r_, i_,k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,3],{r,0,8},{i,0,r}]]

Formula

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

T(r, 0, 3) = A006632(r + 1).

T(r, r, 3) = A000272(r + 1).