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A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

%I A185148 #43 Nov 12 2017 03:02:50
%S A185148 1,6,53,587,7572,109027,1705249,28440320,499208817,9134237407,
%T A185148 172976239886,3371587949969,67351686970929,1374179898145980,
%U A185148 28557595591148315,603118526483125869,12920388129877471030,280324904918707937001,6151595155000424589327,136384555249451824930126
%N A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.
%C A185148 a(n) counts a subset of A025035(n).
%C A185148 a(n) counts a more general set than A005789(n).
%C A185148 a(n) is also the number of (3*n-1)-step walks on 3-dimensional cubic lattice from (1,0,0) to (n,n,n) with steps in {(1,0,0), (0,1,0), (0,0,1)} such that for each point (x,y,z) we have x<=y<=z or x>=y>=z. - _Alois P. Heinz_, Feb 29 2012
%H A185148 Alois P. Heinz and Vaclav Kotesovec, <a href="/A185148/b185148.txt">Table of n, a(n) for n = 1..700</a> (terms 0..200 from Alois P. Heinz)
%F A185148 a(n) ~ c * 27^n / n^4, where c = 0.608287207375... . - _Vaclav Kotesovec_, Sep 03 2014, updated Sep 07 2016
%e A185148 For n = 2 the a(2) = 6 arrangements are:
%e A185148 +---+  +---+  +---+  +---+  +---+  +---+
%e A185148 |1 4|  |1 6|  |1 3|  |1 3|  |1 2|  |1 2|
%e A185148 |2 5|  |2 5|  |2 5|  |2 4|  |3 5|  |3 4|
%e A185148 |3 6|  |3 4|  |4 6|  |5 6|  |4 6|  |5 6|
%e A185148 +---+  +---+  +---+  +---+  +---+  +---+
%e A185148 Only the second of these arrangements is not counted by A005789(2).
%p A185148 b:= proc(x, y, z) option remember;
%p A185148       `if`(x=z, `if`(x=0, 1, 2*b(x-1, y, z)), `if`(x>0, b(x-1, y, z), 0)+
%p A185148       `if`(y>x, b(x, y-1, z), 0)+ `if`(z>y, b(x, y, z-1), 0))
%p A185148     end:
%p A185148 a:= n-> b(n-1, n$2):
%p A185148 seq(a(n), n=1..30);  # _Alois P. Heinz_, Feb 29 2012
%t A185148 b[x_, y_, z_] := b[x, y, z] = If[x == z, If[x == 0, 1, 2*b[x - 1, y, z]], If[x > 0, b[x - 1, y, z], 0] + If[y > x, b[x, y - 1, z], 0] + If[z > y, b[x, y, z - 1], 0]];
%t A185148 a[n_] := b[n - 1, n, n];
%t A185148 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Nov 12 2017, after _Alois P. Heinz_ *)
%Y A185148 Cf. A025035, A005789.
%Y A185148 Column k=3 of A208615. - _Alois P. Heinz_, Feb 29 2012
%K A185148 nonn
%O A185148 1,2
%A A185148 _Olivier Gérard_, Feb 15 2011
%E A185148 More terms and example from _Alois P. Heinz_, Feb 22 2011
%E A185148 Extended beyond a(8) by _Alois P. Heinz_, Feb 22 2012