A379613 a(n) = n^(n - 1) - 2*(n + 1)^(n - 2), by convention a(0) = 0.

0, 0, 0, 1, 14, 193, 2974, 52113, 1034270, 23046721, 571282238, 15617863897, 467291386990, 15198954783153, 534222097472894, 20185726770649633, 816165851488045118, 35167910642711951617, 1609028732603454196606, 77912950297911241532841, 3981118415206568940420878

Offset: 0

Author

Peter Luschny, Dec 27 2024

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385
• 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.

Crossrefs

Cf. A000169, A007334, A374756, A379611.

Programs

Magma

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

Maple

a := n -> ifelse(n = 0, 0, n^(n-1) - 2*(n+1)^(n-2)): seq(a(n), n = 0..20);

Mathematica

{0}~Join~Table[n^(n - 1) - 2*(n + 1)^(n - 2), {n, 20}] (* Michael De Vlieger, Dec 27 2024 *)

SageMath

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

Formula

a(n) = A000169(n) - A007334(n+1) for n > 0. In the context of parking functions this is the difference between the main diagonals of A374756 and A379611. See corollary 3.1 and Table 2 in Butler et al.

E.g.f.: (1/(4*x))*((2*W(-x) + 2 - x)^2 - (4 - 12*x + x^2)), W(x) = Lambert W function. - G. C. Greubel, Mar 19 2025