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A386015 Number of parking functions of size n with a descent in the first position.

%I A386015 #29 Aug 19 2025 16:36:13
%S A386015 0,1,6,50,540,7203,114688,2125764,45000000,1071794405,28378791936,
%T A386015 827150951094,26322173602816,908224365234375,33776997205278720,
%U A386015 1347022612475207432,57346445240058396672,2595972722108590509129,124518400000000000000000,6308807923967155297895610,336682260736692839281065984
%N A386015 Number of parking functions of size n with a descent in the first position.
%H A386015 Paolo Xausa, <a href="/A386015/b386015.txt">Table of n, a(n) for n = 1..350</a>
%H A386015 Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, <a href="https://arxiv.org/abs/2508.11587">Inversions in parking functions</a>, arXiv:2508.11587 [math.CO], 2025.
%H A386015 Wikipedia, <a href="https://en.wikipedia.org/wiki/Parking_function">Parking function</a>.
%F A386015 a(n) = (n/2)*(n+1)^(n-2) for n >= 2.
%F A386015 a(n) = A085389(n) / 2 for n >= 2.
%e A386015 For n=1 the only parking function is 1 and it does not have a descent in the first position so a(1)=0.
%e A386015 For n=2 there are 3 parking functions: 11,12,21. Among them only 21 has a descent in the first position so a(2)=1.
%e A386015 For n=3 there are 16 parking functions: 111,112,121,211,122,212,221,113,131,311,123,132,213,231,312,321. Of these, 211,212,311,213,312, and 321 have a descent in the first position so a(3)=6.
%t A386015 A386015[n_] := If[n == 1, 0, n*(n+1)^(n-2)/2];
%t A386015 Array[A386015, 25] (* _Paolo Xausa_, Aug 07 2025 *)
%Y A386015 Cf. A000272, A085389.
%K A386015 nonn,easy
%O A386015 1,3
%A A386015 _Gabe Udell_, Jul 14 2025