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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.

%I A227579 #21 Jan 03 2021 09:24:59
%S A227579 1,1,5,290,456033,36470203156,237791136700913751,
%T A227579 184570140930218389159747070,23408169635197679203800470649923362577,
%U A227579 637028433009539403532335279417025047587902906655768,4725612998324981086891784010767387049970117446517002416810380479702
%N 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.
%H A227579 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 A227579 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 A227579 b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
%p A227579       i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
%p A227579     end:
%p A227579 a:= n-> `if`(n=0, 1, b([n$n])):
%p A227579 seq(a(n), n=0..9);
%t A227579 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 A227579 a[n_] := If[n == 0, 1, b[Table[n, {n}]]];
%t A227579 a /@ Range[0, 9] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)
%Y A227579 Main diagonal of A227578.
%K A227579 nonn,hard
%O A227579 0,3
%A A227579 _Alois P. Heinz_, Jul 16 2013