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A225111 Number of lattice paths without interior points from {n}^n to {0}^n using steps that decrement one component by 1.

%I A225111 #13 Apr 28 2022 07:56:56
%S A225111 1,1,2,384,8059800,38606650125120,71646205399259162031360,
%T A225111 78385944219935192681549282987212800,
%U A225111 71605824043564034004713155518007394441060661360000,73473023853389304132357517396557811159635782691183541179936000000000
%N A225111 Number of lattice paths without interior points from {n}^n to {0}^n using steps that decrement one component by 1.
%C A225111 An interior point p = (p_1, ..., p_n) has n>0 components with 0<p_i<n for 1<=i<=n.
%H A225111 Alois P. Heinz, <a href="/A225111/b225111.txt">Table of n, a(n) for n = 0..12</a>
%p A225111 b:= proc(n, l) option remember; local m; m:= nops(l);
%p A225111       `if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m]<n, 0,
%p A225111        add(`if`(l[i]=0, 0, b(n, sort(subsop(i=l[i]-1, l)))), i=1..m)))
%p A225111     end:
%p A225111 a:= n-> b(n, [n$n]):
%p A225111 seq(a(n), n=0..9);
%t A225111 b[n_, l_] := b[n, l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1,
%t A225111      If[l[[1]] > 0 && l[[m]] < n, 0, Sum[If[l[[i]] == 0, 0, b[n, Sort[
%t A225111      ReplacePart[l, i -> l[[i]] - 1]]]], {i, 1, m}]]]];
%t A225111 a[n_] := b[n, Array[n&, n]];
%t A225111 Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, Apr 28 2022, after _Alois P. Heinz_ *)
%Y A225111 Main diagonal of A225094.
%K A225111 nonn
%O A225111 0,3
%A A225111 _Alois P. Heinz_, Apr 28 2013