A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.

1, 4, 1, 26, 12, 3, 204, 136, 64, 16, 1771, 1540, 1050, 500, 125, 16380, 17550, 15600, 10800, 5184, 1296, 158224, 201376, 220255, 198940, 139258, 67228, 16807, 1577532, 2324784, 3015936, 3351040, 3063808, 2162688, 1048576, 262144, 16112057, 26978328, 40467492, 53298648, 59960979, 55348596, 39326634, 19131876, 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  |    4      1
2  |   26     12      3
3  |  204    136     64     16
4  | 1771   1540   1050    500    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, A007318, A118971, A329057, A329058, A329059, A329123 (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,4],{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, 4) = A118971(r).

T(r, r, 4) = A000272(r + 1).