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A227584 Number of lattice paths from {4}^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 A227584 #18 Dec 20 2020 08:14:39
%S A227584 1,8,185,7680,456033,34426812,3086989927,315051017342,35566911169298,
%T A227584 4353511908566248,569413385415535738,78713723425497511522,
%U A227584 11403561640157735499129,1719932910431380877877228,268627766543783314569921051,43259068400832620021992394382
%N A227584 Number of lattice paths from {4}^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 A227584 Alois P. Heinz and Vaclav Kotesovec, <a href="/A227584/b227584.txt">Table of n, a(n) for n = 0..422</a> (terms n=51..98 from Vaclav Kotesovec)
%F A227584 From _Vaclav Kotesovec_, Nov 18 2016: (Start)
%F A227584 Recurrence: 2*(n+2)^3*(n+3)*(3*n + 4)*(3*n + 5)*(21546*n^6 - 45513*n^5 - 3699*n^4 + 13101*n^3 + 6745*n^2 + 2032*n + 228)*a(n) = (n+2)*(100641366*n^11 + 49968333*n^10 - 359008281*n^9 - 154345878*n^8 + 328506480*n^7 - 50516019*n^6 - 255412581*n^5 + 41482508*n^4 + 116397064*n^3 + 50875696*n^2 + 13946672*n + 1423680)*a(n-1) - 2*(173359116*n^12 + 510099768*n^11 - 127771911*n^10 - 1779706188*n^9 - 1320942528*n^8 + 1504634418*n^7 + 2094287811*n^6 + 74489810*n^5 - 783633768*n^4 - 274711936*n^3 + 13818000*n^2 + 12588128*n + 862080)*a(n-2) - 16*(2*n - 3)*(2*n + 1)*(3*n - 2)*(3*n - 1)*(4*n + 1)*(4*n + 3)*(21546*n^6 + 83763*n^5 + 91926*n^4 - 25905*n^3 - 108086*n^2 - 58260*n - 5560)*a(n-3).
%F A227584 a(n) ~ 2^(8*n+51/2) / (3^11 * Pi^(3/2) * n^(15/2)).
%F A227584 (End)
%e A227584 a(1) = 8: [(4),(0)], [(4),(1),(0)], [(4),(2),(0)], [(4),(2),(1),(0)], [(4),(3),(0)], [(4),(3),(1),(0)], [(4),(3),(2),(0)], [(4),(3),(2),(1),(0)].
%p A227584 b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
%p A227584       i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
%p A227584     end:
%p A227584 a:= n-> `if`(n=0, 1, b([4$n])):
%p A227584 seq(a(n), n=0..16);
%t A227584 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 A227584 a[n_] := If[n == 0, 1, b[Array[4&, n]]];
%t A227584 a /@ Range[0, 16] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A227584 Row n=4 of A227578.
%K A227584 nonn
%O A227584 0,2
%A A227584 _Alois P. Heinz_, Jul 16 2013