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A265130 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of k into distinct parts with largest part n and n<=k<=n*(n+1)/2.

%I A265130 #14 Aug 22 2021 09:25:01
%S A265130 1,1,5,32,272,2957,39531,629806,11673074,247028567,5881190801,
%T A265130 155651692748,4534744862052,144246963009697,4975152075900887,
%U A265130 184958685188293274,7373625038400716198,313817002976857310507,14201832585602869616349,681022860320979979626232
%N A265130 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of k into distinct parts with largest part n and n<=k<=n*(n+1)/2.
%H A265130 R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011
%p A265130 p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
%p A265130          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
%p A265130 g:= (n, i, l)-> `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l),
%p A265130    g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):
%p A265130 a:= n-> `if`(n=0, 1, add(g(k-n, n-1, [n]), k=n..n*(n+1)/2)):
%p A265130 seq(a(n), n=0..10);
%t A265130 p[l_] := Function[n, n!*Det[Table[Function [t,
%t A265130      If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]][Length[l]];
%t A265130 g[n_, i_, l_] := If[i(i+1)/2 < n, 0,
%t A265130      If[n == 0, p[l], g[n, i - 1, l] +
%t A265130      If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
%t A265130 a[n_] := If[n == 0, 1, Sum[g[k - n, n - 1, {n}], {k, n, n(n+1)/2}]];
%t A265130 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Aug 22 2021, after _Alois P. Heinz_ *)
%Y A265130 Column sums of A265018, A265019.
%K A265130 nonn
%O A265130 0,3
%A A265130 _Alois P. Heinz_, Dec 02 2015