cp's OEIS Frontend

A265007 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n.

%I A265007 #17 Aug 22 2021 09:24:51
%S A265007 1,1,3,7,18,40,97,216,499,1112,2502,5503,12197,26582,58088,125619,
%T A265007 271713,583228,1251115,2668651,5685053,12059993,25544291,53926003,
%U A265007 113666195,238946232,501546514,1050430420,2196869731,4586021745,9560876381,19900839742,41373446190
%N A265007 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n.
%H A265007 Alois P. Heinz, <a href="/A265007/b265007.txt">Table of n, a(n) for n = 0..50</a>
%H A265007 R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011
%e A265007 The number of lambda-parking functions induced by the partitions of 4:
%e A265007 1 by [1,1,1,1]: [1,1,1,1],
%e A265007 4 by [1,1,2]: [1,1,1], [1,1,2], [1,2,1], [2,1,1],
%e A265007 4 by [2,2]: [1,1], [1,2], [2,1], [2,2],
%e A265007 5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],
%e A265007 4 by [4]: [1], [2], [3], [4].
%e A265007 a(4) = 1 + 4 + 4 + 5 + 4 = 18.
%p A265007 p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
%p A265007          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
%p A265007 g:= (n, i, l)-> `if`(n=0 or i=1, p([1$n, l[]]), g(n, i-1, l)
%p A265007                +`if`(i>n, 0, g(n-i, i, [i, l[]]))):
%p A265007 a:= n-> g(n$2, []):
%p A265007 seq(a(n), n=0..20);
%t A265007 p[l_] := With[{n = Length[l]}, n! Det[Table[With[{t = j - i + 1},
%t A265007      If[t < 0, 0, l[[i]]^t/t!]], {i, n}, {j, n}]]];
%t A265007 g[n_, i_, l_] := If[n == 0 || i == 1, p[Join[
%t A265007      Table[1, {n}], l]], g[n, i - 1, l] +
%t A265007      If[i > n, 0, g[n - i, i, Prepend[l, i]]]];
%t A265007 a[n_] := If[n == 0, 1, g[n, n, {}]];
%t A265007 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Aug 22 2021, after _Alois P. Heinz_ *)
%Y A265007 Cf. A000041, A255047, A265016.
%K A265007 nonn
%O A265007 0,3
%A A265007 _Alois P. Heinz_, Nov 29 2015