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A265016 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n into distinct parts.

%I A265016 #22 Aug 20 2021 04:26:59
%S A265016 1,1,2,6,9,20,43,74,130,241,493,774,1413,2286,3987,7287,11650,19235,
%T A265016 31581,50852,80867,141615,214538,349179,541603,859759,1303221,2054700,
%U A265016 3277493,4960397,7652897,11662457,17703655,26603187,40043433,59384901,92234897,134538472
%N A265016 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n into distinct parts.
%H A265016 Alois P. Heinz, <a href="/A265016/b265016.txt">Table of n, a(n) for n = 0..100</a>
%H A265016 Richard P. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011.
%e A265016 The number of lambda-parking functions induced by the partitions of 4 into distinct parts:
%e A265016 5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],
%e A265016 4 by [4]: [1], [2], [3], [4].
%e A265016 a(4) = 5 + 4 = 9.
%p A265016 p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
%p A265016          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
%p A265016 g:= (n, i, l)->  `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l),
%p A265016     g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):
%p A265016 a:= n-> g(n$2, []):
%p A265016 seq(a(n), n=0..35);
%t A265016 p[l_] := With[{n = Length[l]}, n!*Det[Table[Function[t,
%t A265016      If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]];
%t A265016 g[n_, i_, l_] := If[i (i + 1)/2 < n, 0, If[n == 0, p[l],
%t A265016      g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
%t A265016 a[n_] := If[n == 0, 1, g[n, n, {}]];
%t A265016 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Aug 20 2021, after _Alois P. Heinz_ *)
%Y A265016 Row sums of A265017, A265018, A265019, A265020.
%Y A265016 Cf. A000009, A265007, A265202.
%K A265016 nonn
%O A265016 0,3
%A A265016 _Alois P. Heinz_, Nov 30 2015