A372842
a(n) is the number of parking functions of order n for which the second spot is lucky.
Original entry on oeis.org
2, 11, 87, 908, 11824, 184944, 3381341, 70805696, 1671605646, 43938023168, 1272792377875, 40291409169408, 1383927524621468, 51265193822056448, 2037343816037147001, 86467962304018300928, 3903480077867017448410, 186771397981175865606144, 9441767566333191196904591
Offset: 2
For clarity, we write parentheses around parking functions. For n = 2, the a(2) = 2 solutions are the parking functions of length 2 with a lucky second spot are (1,2) and (2,1). The parking function (1,1) is not one of the solutions because the car which parks in the second spot did not prefer the second spot; this parking function does not contribute to our count.
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Array[(3/4)*(# + 1)^(# - 1) - (1/4)*(# - 1)^(# - 1) &, 19, 2] (* Michael De Vlieger, Jun 26 2024 *)
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def A372842(n): return 3*(n+1)**(n-1)-(n-1)**(n-1)>>2 # Chai Wah Wu, Jun 26 2024
A372844
a(n) is the number of parking functions of order n for which the fourth spot is lucky.
Original entry on oeis.org
64, 708, 9421, 148992, 2742090, 57671104, 1365730231, 35980443648, 1044117402868, 33098695234560, 1138160856018369, 42200676331159552, 1678427133899138494, 71282668099352051712, 3219814814790580711915, 154137012617228775849984, 7795444201708762192584744, 415337944634097426474729472
Offset: 4
For clarity, we write parentheses around parking functions. For n = 4, there are a(4) = 64 solutions. An example of a parking function of order 4 with a lucky fourth spot is (1,4,2,2); here, the second car parks in the fourth spot which is its preferred spot. This parking function contributes to our count. A non-example is the parking function (1,2,1,2); here, the last car parks in the fourth spot, but its preference is spot 2. This parking function does not contribute to our count.
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a[n_]:=(5/8)*(n+1)^(n-1)-(1/8)*(13*n^2-26*n+9)*(n-3)^(n-3); Array[a,19,4] (* Stefano Spezia, Jun 26 2024 *)
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def A372844(n): return 5*(n+1)**(n-1)-(13*(n-1)**2-4)*(n-3)**(n-3)>>3 # Chai Wah Wu, Jun 26 2024
A372845
a(n) is the number of parking functions of order n for which the fifth spot is lucky.
Original entry on oeis.org
625, 8733, 140298, 2600879, 54921875, 1303885965, 34409008596, 999711522899, 31719176377701, 1091467041015625, 40491113522829630, 1611131116280526327, 68448950529246552887, 3092734133786108912869, 148090628302001953125000, 7491257174986774088059995, 399205026805287676036911049
Offset: 5
For clarity, we write parentheses around parking functions. For n = 6, there are a(6) = 8733 solutions. An example of a parking function of order 6 with a lucky fifth spot is (1,4,1,5,2,2); here, the fourth car parks in the fifth spot which is its preferred spot. This parking function contributes to our count. A non-example is the parking function (1,1,1,1,1,5); here, the fifth car parks in the fifth spot, but its preference is spot 1. This parking function does not contribute to our count.
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a[n_]:=(3/5)*(n+1)^(n-1)-(1/30)*(118*n^3-531*n^2+659*n-192)*(n-4)^(n-4); Array[a,17,5] (* Stefano Spezia, Jun 26 2024 *)
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def A372845(n): return (18*(n+1)**(n-1)-(n*(n*(59*((n<<1)-9))+659)-192)*(n-4)**(n-4))//30 # Chai Wah Wu, Jun 26 2024
A374756
Triangle read by rows: T(n,k) is the number of parking functions of order n where the k-th car is lucky.
Original entry on oeis.org
1, 3, 2, 16, 11, 9, 125, 87, 74, 64, 1296, 908, 783, 708, 625, 16807, 11824, 10266, 9421, 8733, 7776, 262144, 184944, 161221, 148992, 140298, 131632, 4782969, 3381341, 2955366, 2742090, 2600879, 2480787, 100000000, 70805696, 61999923, 57671104, 54921875, 52779840, 2357947691, 1671605646, 1465709426, 1365730231, 1303885965, 1258181726
Offset: 1
Triangle begins:
1;
3, 2;
16, 11, 9;
125, 87, 74, 64;
1296, 908, 783, 708, 625;
16807, 11824, 10266, 9421, 8733, 7776;
...
For clarity, we write parentheses around parking functions. For n = 3 and k = n-1 = 2, the T(3,2) = 11 solutions are the parking functions of length 3 with a lucky second spot: (1,2,1),(1,2,2),(1,2,3),(1,3,2),(2,1,1),(2,1,2),(2,1,3),(2,2,1),(2,3,1),(3,1,2),(3,2,1). There are 5 parking functions of length 3 which do not have a lucky second spot: (1,1,1),(1,1,2),(1,1,3),(1,3,1),(3,1,1). For all of these, the car which parks in the second spot did not prefer the second spot; these parking functions do not contribute to our count.
Showing 1-4 of 4 results.
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