A375616
a(n) is the number of lucky cars in all parking functions of order n.
Original entry on oeis.org
0, 1, 5, 36, 350, 4320, 64827, 1146880, 23383404, 540000000, 13933327265, 397303087104, 12407264266410, 421154777645056, 15439814208984375, 607985949695016960, 25593429637028941208, 1146928904801167933440, 54515427164280400691709, 2739404800000000000000000
Offset: 0
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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:
a:= n-> add(k*coeff(b(n), x, k), k=1..n):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 21 2024
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b[n_] := b[n] = If[n == 0, 1, Expand[x*Product[(n+1-k) + k*x, {k, 2, n}]]];
a[n_] := Sum[k*Coefficient[b[n], x, k], {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 31 2024, after Alois P. Heinz *)
A379611
Table read by rows: T(n, k) = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2), by convention T(1, 0) = 1.
Original entry on oeis.org
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
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;
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T := (n, k) -> ifelse(n=1 and k=0, 1, (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2)):
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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 *)
A379613
a(n) = n^(n - 1) - 2*(n + 1)^(n - 2), by convention a(0) = 0.
Original entry on oeis.org
0, 0, 0, 1, 14, 193, 2974, 52113, 1034270, 23046721, 571282238, 15617863897, 467291386990, 15198954783153, 534222097472894, 20185726770649633, 816165851488045118, 35167910642711951617, 1609028732603454196606, 77912950297911241532841, 3981118415206568940420878
Offset: 0
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A379613:= func< n | n eq 0 select 0 else n^(n-1) -2*(n+1)^(n-2) >;
[A379613(n): n in [0..30]]; // G. C. Greubel, Mar 19 2025
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a := n -> ifelse(n = 0, 0, n^(n-1) - 2*(n+1)^(n-2)): seq(a(n), n = 0..20);
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{0}~Join~Table[n^(n - 1) - 2*(n + 1)^(n - 2), {n, 20}] (* Michael De Vlieger, Dec 27 2024 *)
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def A379613(n): return 0 if n==0 else n^(n-1) -2*(n+1)^(n-2)
print([A379613(n) for n in range(31)]) # G. C. Greubel, Mar 19 2025
A374533
a(n) is the number of parking functions of order n where the (n-1)-st spot is lucky.
Original entry on oeis.org
3, 11, 74, 708, 8733, 131632, 2342820, 48068672, 1116809255
Offset: 2
For clarity, we write parentheses around parking functions. For n = 3, the a(3) = 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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