A227602 Number of lattice paths from {5}^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, 16, 1257, 238636, 77767945, 36470203156, 22228291051255, 16513520723284922, 14323116388173517180, 14071120934043157192832, 15313737501505148093502344, 18156604289232210133044514152, 23151467541948649805794187113781, 31425801906523386705389663813716908

Offset: 0

Author

Alois P. Heinz, Jul 17 2013

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..60

Crossrefs

Row n=5 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([5$n])):
seq(a(n), n=0..14);

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[Array[5&, n]]];
a /@ Range[0, 14] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

a(n) ~ 9 * 5^(5*n + 41/2) / (2^37 * Pi^2 * n^12). - Vaclav Kotesovec, Nov 21 2016