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A210486 Number of paths starting at {3}^n to a border position where one component equals 0 using steps that decrement one component by 1.

%I A210486 #19 Mar 29 2017 12:10:49
%S A210486 0,1,20,543,22096,1304045,106478916,11545342795,1608000044288,
%T A210486 280061940550041,59677171216017940,15278632095285640631,
%U A210486 4628964787172536267920,1638318264614752659427333,669895681115518466689138436,313418973409285344224352078435
%N A210486 Number of paths starting at {3}^n to a border position where one component equals 0 using steps that decrement one component by 1.
%H A210486 Alois P. Heinz, <a href="/A210486/b210486.txt">Table of n, a(n) for n = 0..237</a>
%F A210486 a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n+3/2) / exp(2*n-1). - _Vaclav Kotesovec_, Sep 02 2014
%e A210486 a(1) = 1: [3, 2, 1, 0].
%e A210486 a(2) = 20: [33, 23, 13, 03], [33, 23, 13, 12, 02], [33, 23, 13, 12, 11, 01], [33, 23, 13, 12, 11, 10], [33, 23, 22, 12, 02], [33, 23, 22, 12, 11, 01], [33, 23, 22, 12, 11, 10], [33, 23, 22, 21, 11, 01], [33, 23, 22, 21, 11, 10], [33, 23, 22, 21, 20], [33, 32, 22, 12, 02], [33, 32, 22, 12, 11, 01], [33, 32, 22, 12, 11, 10], [33, 32, 22, 21, 11, 01], [33, 32, 22, 21, 11, 10], [33, 32, 22, 21, 20], [33, 32, 31, 21, 11, 01], [33, 32, 31, 21, 11, 10], [33, 32, 31, 21, 20], [33, 32, 31, 30].
%p A210486 a:= proc(n) option remember; `if`(n<3, [0, 1, 20][n+1],
%p A210486       ((n-1)*(n-2)*(n+1)*a(n-3) -(n-1)*(3*n^2-2*n-4)*a(n-2)
%p A210486       +(2*n+1)*(n^2-n+2)*a(n-1)) / (n-1))
%p A210486     end:
%p A210486 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 23 2013
%t A210486 a[n_] := a[n] = If[n<3, {0, 1, 20}[[n+1]], ((n-1)*(n-2)*(n+1)*a[n-3] - (n-1)*(3*n^2 - 2*n - 4)*a[n-2] + (2*n+1)*(n^2 - n + 2)*a[n-1]) / (n-1)];
%t A210486 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 29 2017, after _Alois P. Heinz_ *)
%Y A210486 Row n=3 of A210472.
%K A210486 nonn,walk
%O A210486 0,3
%A A210486 _Alois P. Heinz_, Jan 23 2013