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A227596 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one component such that for each point (p_1,p_2,...,p_5) we have p_1<=p_2<=...<=p_5.

%I A227596 #15 Dec 20 2020 07:48:43
%S A227596 1,1,132,49100,34426812,36470203156,51630369256916,90650832149396184,
%T A227596 187978502469162658572,445073778727031182727610,
%U A227596 1174228543974568589770758656,3389786811049267225428045061056,10559434352810002520295112134863908
%N A227596 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one component such that for each point (p_1,p_2,...,p_5) we have p_1<=p_2<=...<=p_5.
%H A227596 Vaclav Kotesovec, <a href="/A227596/b227596.txt">Table of n, a(n) for n = 0..116</a>
%F A227596 a(n) ~ 2*sqrt(5) * 6^(5*n+21) / (5^7 * 7^12 * Pi^2 * n^12). - _Vaclav Kotesovec_, Nov 20 2016
%p A227596 b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
%p A227596       i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
%p A227596     end:
%p A227596 a:= n-> `if`(n=0, 1, b([n$5])):
%p A227596 seq(a(n), n=0..13);
%t A227596 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 A227596 a[n_] := If[n == 0, 1, b[Array[n&, 5]]];
%t A227596 a /@ Range[0, 13] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A227596 Column k=5 of A227578.
%K A227596 nonn
%O A227596 0,3
%A A227596 _Alois P. Heinz_, Jul 17 2013