A379611 Table read by rows: T(n, k) = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2), by convention T(1, 0) = 1.
2, 1, 1, 4, 3, 2, 20, 16, 12, 8, 150, 125, 100, 75, 50, 1512, 1296, 1080, 864, 648, 432, 19208, 16807, 14406, 12005, 9604, 7203, 4802, 294912, 262144, 229376, 196608, 163840, 131072, 98304, 65536, 5314410, 4782969, 4251528, 3720087, 3188646, 2657205, 2125764, 1594323, 1062882
Offset: 0
Examples
Triangle starts: [0] 2; [1] 1, 1; [2] 4, 3, 2; [3] 20, 16, 12, 8; [4] 150, 125, 100, 75, 50; [5] 1512, 1296, 1080, 864, 648, 432; [6] 19208, 16807, 14406, 12005, 9604, 7203, 4802; [7] 294912, 262144, 229376, 196608, 163840, 131072, 98304, 65536;
Links
- Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 7, corollary 3.1.
Programs
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Maple
T := (n, k) -> ifelse(n=1 and k=0, 1, (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2)):
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Mathematica
T[n_, k_] := T[n, k] = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2); T[1, 0] := 1; Flatten@ Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Michael De Vlieger, Dec 27 2024 *)
Formula
T(n, k) = (n + 1)^(n - 2)*(n - k + 2), if (n, k) != (1, 0).
T(n, k) = (1 - (k - 1)/(n + 1))*(n + 1)^(n - 1), if (n, k) != (1, 0).