A227579 - OEIS

%I #21 Jan 03 2021 09:24:59

%S 1,1,5,290,456033,36470203156,237791136700913751,

%T 184570140930218389159747070,23408169635197679203800470649923362577,

%U 637028433009539403532335279417025047587902906655768,4725612998324981086891784010767387049970117446517002416810380479702

%N 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.

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A227579/b227579.txt">Table of n, a(n) for n = 0..14</a> (terms 0..12 from Alois P. Heinz)

%e 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)].

%p b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(

%p       i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))

%p     end:

%p a:= n-> `if`(n=0, 1, b([n$n])):

%p seq(a(n), n=0..9);

%t 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]}]];

%t a[n_] := If[n == 0, 1, b[Table[n, {n}]]];

%t a /@ Range[0, 9] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)

%Y Main diagonal of A227578.

%K nonn,hard

%O 0,3

%A _Alois P. Heinz_, Jul 16 2013