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A265202 Total number of lambda-parking functions induced by all partitions of n into distinct parts.

%I A265202 #20 Jun 25 2025 20:26:20
%S A265202 1,1,2,6,9,15,36,53,78,119,286,401,591,829,1232,2910,4084,5789,8070,
%T A265202 11281,15823,37747,51622,72919,98986,136600,181648,254638,586891,
%U A265202 799841,1110303,1495279,2018749,2657612,3552560,4738775,10857521,14560375,20061359,26603227
%N A265202 Total number of lambda-parking functions induced by all partitions of n into distinct parts.
%H A265202 Alois P. Heinz, <a href="/A265202/b265202.txt">Table of n, a(n) for n = 0..400</a>
%H A265202 Richard P. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011
%e A265202 a(0) = 1: [].
%e A265202 a(1) = 1: [1].
%e A265202 a(2) = 2: [1], [2].
%e A265202 a(3) = 6: [1], [2], [3], [1,1], [1,2], [2,1].
%e A265202 a(4) = 9: [1], [2], [3], [4], [1,1], [1,2], [1,3], [2,1], [3,1].
%e A265202 a(5) = 15: [1], [2], [3], [4], [5], [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1].
%e A265202 a(6) = 36: [1], [2], [3], [4], [5], [6], [1,1], [1,2], [1,3], [1,4], [1,5], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [4,1], [4,2], [5,1], [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,3,1], [3,1,1], [3,1,2], [3,2,1].
%p A265202 b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!)+
%p A265202       `if`(n<t, 0, add(b(p+1, `if`(i=j, g+1, 1), n-max(j, t), j,
%p A265202        max(j+1, t+1))/`if`(i=j, 1, g!), j=i..n))
%p A265202     end:
%p A265202 a:= n-> `if`(n=0, 1, b(0$2, n, 1$2)):
%p A265202 seq(a(n), n=0..50);
%t A265202 b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!] + If[n<t, 0, Sum[b[p+1, If[i==j, g+1, 1], n-Max[j, t], j, Max[j+1, t+1]]/If[i==j, 1, g!], {j, i, n}]]; a[n_] := If[n==0, 1, b[0, 0, n, 1, 1]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 02 2017, translated from Maple *)
%Y A265202 Row sums of A265208.
%Y A265202 Cf. A000009, A255047, A265016.
%K A265202 nonn
%O A265202 0,3
%A A265202 _Alois P. Heinz_, Dec 04 2015