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A375616 a(n) is the number of lucky cars in all parking functions of order n.

%I A375616 #27 Dec 27 2024 08:55:16
%S A375616 0,1,5,36,350,4320,64827,1146880,23383404,540000000,13933327265,
%T A375616 397303087104,12407264266410,421154777645056,15439814208984375,
%U A375616 607985949695016960,25593429637028941208,1146928904801167933440,54515427164280400691709,2739404800000000000000000
%N A375616 a(n) is the number of lucky cars in all parking functions of order n.
%C A375616 This sequence enumerates lucky cars in parking functions of order n (where a lucky spot is one which is parked in by a car which prefers that spot).
%H A375616 Alois P. Heinz, <a href="/A375616/b375616.txt">Table of n, a(n) for n = 0..386</a>
%H A375616 Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See p. 10.
%F A375616 a(n) = Sum_{k=1..n} k*A370832(n,k) = Sum_{k=1..n} A374756(n,k).
%p A375616 b:= proc(n) option remember; `if`(n=0, 1,
%p A375616       expand(x*mul((n+1-k)+k*x, k=2..n)))
%p A375616     end:
%p A375616 a:= n-> add(k*coeff(b(n), x, k), k=1..n):
%p A375616 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 21 2024
%t A375616 b[n_] := b[n] = If[n == 0, 1, Expand[x*Product[(n+1-k) + k*x, {k, 2, n}]]];
%t A375616 a[n_] := Sum[k*Coefficient[b[n], x, k], {k, 1, n}];
%t A375616 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 31 2024, after _Alois P. Heinz_ *)
%Y A375616 Row sums of A374756.
%Y A375616 Cf. A370832.
%K A375616 nonn
%O A375616 0,3
%A A375616 _Kimberly P. Hadaway_, Aug 21 2024, suggested by _Andrew Howroyd_
%E A375616 a(6)-a(19) from _Alois P. Heinz_, Aug 21 2024