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A370832
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Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 0 <= k <= n.
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3
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1, 0, 1, 0, 1, 2, 0, 2, 8, 6, 0, 6, 37, 58, 24, 0, 24, 204, 504, 444, 120, 0, 120, 1318, 4553, 6388, 3708, 720, 0, 720, 9792, 44176, 87296, 81136, 33984, 5040, 0, 5040, 82332, 463860, 1203921, 1582236, 1064124, 341136, 40320, 0, 40320, 773280, 5270480, 17164320, 29724000, 28328480, 14602320, 3733920, 362880
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OFFSET
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0,6
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COMMENTS
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A car is called "lucky" if it gets its preferred parking spot.
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LINKS
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FORMULA
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T(n, n) = n!.
T(n, 1) = (n-1)!.
Sum_{k=1..n} T(n, k) = (n+1)^(n-1).
G.f. for row n>0: x * Product_{j=2..n} (n + 1 + j*(x-1)).
T(n, k) = [x^k] (x*(x - 1)^n*Pochhammer((n + x) / (x - 1), n)) / (n + x). - Peter Luschny, Jun 27 2024
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EXAMPLE
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Table begins:
n\k| 0 1 2 3 4 5 6 7 8
---+-------------------------------------------------------------
0 | 1
1 | 0 1
2 | 0 1 2
3 | 0 2 8 6
4 | 0 6 37 58 24
5 | 0 24 204 504 444 120
6 | 0 120 1318 4553 6388 3708 720
7 | 0 720 9792 44176 87296 81136 33984 5040
8 | 0 5040 82332 463860 1203921 1582236 1064124 341136 40320
...
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1,
expand(x*mul((n+1-k)+k*x, k=2..n)))
end:
T:= (n, k)-> coeff(b(n), x, k):
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MATHEMATICA
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row[n_] := (x (x - 1)^n Pochhammer[(n + x) / (x - 1), n]) / (n + x);
Table[CoefficientList[Series[row[n], {x, 0, n}], x], {n, 0, 8}] // Flatten
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CROSSREFS
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Cf. A000142 (main diagonal and column k=1 shifted).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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