A227579 Number of lattice paths from {n}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

1, 1, 5, 290, 456033, 36470203156, 237791136700913751, 184570140930218389159747070, 23408169635197679203800470649923362577, 637028433009539403532335279417025047587902906655768, 4725612998324981086891784010767387049970117446517002416810380479702

Offset: 0

Author

Alois P. Heinz, Jul 16 2013

Keywords

• nonn
• hard

Examples

			a(2) = 5: [(2,2),(0,2),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)].
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..14 (terms 0..12 from Alois P. Heinz)

Crossrefs

Main diagonal of A227578.

Programs

Maple

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$n])):
seq(a(n), n=0..9);

Mathematica

b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i - 1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := If[n == 0, 1, b[Table[n, {n}]]];
a /@ Range[0, 9] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)