A265130 - OEIS

%I #14 Aug 22 2021 09:25:01

%S 1,1,5,32,272,2957,39531,629806,11673074,247028567,5881190801,

%T 155651692748,4534744862052,144246963009697,4975152075900887,

%U 184958685188293274,7373625038400716198,313817002976857310507,14201832585602869616349,681022860320979979626232

%N 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 R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011

%p p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)

%p          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):

%p g:= (n, i, l)-> `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l),

%p    g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):

%p a:= n-> `if`(n=0, 1, add(g(k-n, n-1, [n]), k=n..n*(n+1)/2)):

%p seq(a(n), n=0..10);

%t p[l_] := Function[n, n!*Det[Table[Function [t,

%t      If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]][Length[l]];

%t g[n_, i_, l_] := If[i(i+1)/2 < n, 0,

%t      If[n == 0, p[l], g[n, i - 1, l] +

%t      If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];

%t a[n_] := If[n == 0, 1, Sum[g[k - n, n - 1, {n}], {k, n, n(n+1)/2}]];

%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Aug 22 2021, after _Alois P. Heinz_ *)

%Y Column sums of A265018, A265019.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 02 2015