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A227583 Number of lattice paths from {n}^4 to {0}^4 using steps that decrement one component such that for each point (p_1,p_2,...,p_4) we have p_1<=p_2<=...<=p_4.

%I A227583 #21 Dec 20 2020 08:14:10
%S A227583 1,1,42,3532,456033,77767945,16104165970,3848596333400,
%T A227583 1026843977181745,298985252352030713,93462550593036735356,
%U A227583 30988255904733937513266,10797084830552485796491313,3924879717822914648655364113,1479988943331198988162565625314
%N A227583 Number of lattice paths from {n}^4 to {0}^4 using steps that decrement one component such that for each point (p_1,p_2,...,p_4) we have p_1<=p_2<=...<=p_4.
%H A227583 Alois P. Heinz, <a href="/A227583/b227583.txt">Table of n, a(n) for n = 0..200</a>
%F A227583 a(n) ~ 3*5^(4*n+11)/(2^13*Pi^(3/2)*(3*n)^(15/2)). - _Vaclav Kotesovec_, Jul 19 2013
%e A227583 a(0) = 1: [(0,0,0,0)].
%e A227583 a(1) = 1: [(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1),(0,0,0,0)].
%e A227583 a(2) = 42: [(2,2,2,2),(0,2,2,2),(0,0,2,2),(0,0,0,2),(0,0,0,0)], ..., [(2,2,2,2),(1,2,2,2),(1,1,2,2),(1,1,1,2),(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1),(0,0,0,0)].
%p A227583 b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
%p A227583       i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
%p A227583     end:
%p A227583 a:= n-> `if`(n=0, 1, b([n$4])):
%p A227583 seq(a(n), n=0..16);
%p A227583 # second Maple program:
%p A227583 a:= proc(n) option remember; `if`(n<4, [1, 1, 42, 3532][n+1], (
%p A227583       (n+2)*(176594184*n^7-1252171596*n^6+3635870170*n^5
%p A227583       -5627048591*n^4 +4990722596*n^3 -2520013645*n^2+664396290*n
%p A227583       -69572160) *(n+1)^2* a(n-1) -3*(n-2)*(n+1)*(350669592*n^8
%p A227583       -2376848760*n^7+6486604406*n^6-9505222590*n^5
%p A227583       +8466195947*n^4-4746390780*n^3+1536942539*n^2-253906050*n
%p A227583       +14945856) *a(n-2) +(n-3)*(1575354456*n^7-8934392532*n^6
%p A227583       +18759259714*n^5-19551471603*n^4 +12013819684*n^3
%p A227583       -4838385873*n^2+1019903146*n-96686592)*(n-2)^2 *a(n-3)
%p A227583       -5000*(n-4)*(139932*n^4-226356*n^3+87227*n^2-21684*n-1919)
%p A227583       *(n-3)^2 *(n-2)^3 *a(n-4)) / (2*(n+3)*(139932*n^4-786084*n^3
%p A227583       +1605887*n^2-1434934*n+473280)*(n+2)^2*(n+1)^3))
%p A227583     end:
%p A227583 seq(a(n), n=0..20);
%t A227583 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 A227583 a[n_] := b[Array[n&, 4]];
%t A227583 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ in A227578 *)
%Y A227583 Column k=4 of A227578.
%K A227583 nonn
%O A227583 0,3
%A A227583 _Alois P. Heinz_, Jul 16 2013