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A229465 Number of lattice paths from {2}^n to {0}^n using steps that decrement one component or all components by the same positive integer.

%I A229465 #12 Dec 22 2020 03:53:38
%S A229465 1,2,22,248,6506,292442,19450082,1781791202,214899390722,
%T A229465 33007840951682,6290830043769602,1456812593474515202,
%U A229465 402910665233497344002,131173228963457333452802,49656810289226589524275202,21628258853895326260083456002,10739534026001485870629015552002
%N A229465 Number of lattice paths from {2}^n to {0}^n using steps that decrement one component or all components by the same positive integer.
%H A229465 Alois P. Heinz, <a href="/A229465/b229465.txt">Table of n, a(n) for n = 0..200</a>
%F A229465 a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n+1/2) / exp(2*n-1). - _Vaclav Kotesovec_, Jul 16 2014
%p A229465 a:= proc(n) option remember; `if`(n<5, [1, 2, 22, 248, 6506][n+1],
%p A229465      ((64481193996*n^5 -656050382562*n^4 +1835465682464*n^3
%p A229465       -3691825299357*n^2 +10428520019257*n -9978603085078)*a(n-1)
%p A229465       -(64481193996*n^6 -251022627918*n^5 -4253631972584*n^4
%p A229465       +29686486719123*n^3 -71916661134305*n^2 +77149141951487*n
%p A229465       -30090569866279)*a(n-2) +(n-2)*(437268351642*n^5
%p A229465       -5777340617365*n^4 +26203609431616*n^3 -50411340883791*n^2
%p A229465       +38226810988733*n -9795152028455)*a(n-3) -(n-2)*(n-3)*
%p A229465       (170273280324*n^4 -2136687453608*n^3 +8692120865702*n^2
%p A229465       -11643795721897*n +4287224601259)*a(n-4) -(n-6)*(n-2)*(n-3)*
%p A229465       (n-4)*(202513877322*n^2-310611483677*n+98391999767)*a(n-5))/
%p A229465       (32240596998*n^3-328025191281*n^2+768115007074*n-189524735891))
%p A229465     end:
%p A229465 seq(a(n), n=0..20);
%t A229465 b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}], 0] + Sum[Sum[b[Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]];
%t A229465 a[k_] := b[Array[2&, k]];
%t A229465 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ in A229345 *)
%Y A229465 Row n=2 of A229345.
%K A229465 nonn
%O A229465 0,2
%A A229465 _Alois P. Heinz_, Sep 24 2013