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a(n) is the number of lucky cars in all parking functions of order n.
1

%I #24 Aug 31 2024 08:32:57

%S 0,1,5,36,350,4320,64827,1146880,23383404,540000000,13933327265,

%T 397303087104,12407264266410,421154777645056,15439814208984375,

%U 607985949695016960,25593429637028941208,1146928904801167933440,54515427164280400691709,2739404800000000000000000

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

%C 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 Alois P. Heinz, <a href="/A375616/b375616.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) = Sum_{k=1..n} k*A370832(n,k) = Sum_{k=1..n} A374756(n,k).

%p b:= proc(n) option remember; `if`(n=0, 1,

%p expand(x*mul((n+1-k)+k*x, k=2..n)))

%p end:

%p a:= n-> add(k*coeff(b(n), x, k), k=1..n):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 21 2024

%t b[n_] := b[n] = If[n == 0, 1, Expand[x*Product[(n+1-k) + k*x, {k, 2, n}]]];

%t a[n_] := Sum[k*Coefficient[b[n], x, k], {k, 1, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 31 2024, after _Alois P. Heinz_ *)

%Y Row sums of A374756.

%Y Cf. A370832.

%K nonn

%O 0,3

%A _Kimberly P. Hadaway_, Aug 21 2024, suggested by _Andrew Howroyd_

%E a(6)-a(19) from _Alois P. Heinz_, Aug 21 2024