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A214159 Number of n X 3 nonconsecutive tableaux.

%I A214159 #30 Oct 06 2018 19:00:17
%S A214159 1,0,1,6,72,960,14257,228738,3896062,69590192,1291977768,24767357824,
%T A214159 487787117424,9830929279248,202119964701201,4228355224663002,
%U A214159 89820746194643154,1934075860467190608,42153272431384984926,928785495190695948612,20666589509182675178064
%N A214159 Number of n X 3 nonconsecutive tableaux.
%C A214159 A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.
%H A214159 Alois P. Heinz, <a href="/A214159/b214159.txt">Table of n, a(n) for n = 0..700</a>
%H A214159 T. Y. Chow, H. Eriksson and C. K. Fan, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i2a3.html">Chess tableaux</a>, Elect. J. Combin., 11 (2) (2005), #A3.
%H A214159 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A214159 a(n) ~ 3^(3*n+13/2) / (2^12 * Pi * n^4). - _Vaclav Kotesovec_, Jul 16 2014
%e A214159 a(2) = 1:
%e A214159   [1 3 5]
%e A214159   [2 4 6].
%e A214159 a(3) = 6:
%e A214159   [1 4 7]   [1 3 7]   [1 4 6]   [1 3 6]   [1 3 6]   [1 3 5]
%e A214159   [2 5 8]   [2 5 8]   [2 5 8]   [2 5 8]   [2 4 8]   [2 6 8]
%e A214159   [3 6 9]   [4 6 9]   [3 7 9]   [4 7 9]   [5 7 9]   [4 7 9].
%p A214159 b:= proc(l, t) option remember; local n, s; n, s:= nops(l),
%p A214159        add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and l[i]>
%p A214159       `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p A214159     end:
%p A214159 a:= n-> b([3$n], 0):
%p A214159 seq(a(n), n=0..20);
%p A214159 # second Maple program:
%p A214159 a:= proc(n) option remember; `if`(n<3, [1, 0, 1][n+1],
%p A214159       ((22120*n^5 -58226*n^4 +31547*n^3 +42158*n^2 -53589*n +16146)*a(n-1)
%p A214159       -(5988 -27068*n +50360*n^2 -62154*n^3 +49630*n^4 -15400*n^5)*a(n-2)
%p A214159       -(7560*n^5 -23058*n^4 +24333*n^3 -11976*n^2 +2967*n -306)*a(n-3))
%p A214159       /(6*(n+2)*(140*n^2-357*n+234)*(n+1)^2))
%p A214159     end:
%p A214159 seq(a(n), n=0..25);
%t A214159 b[l_, t_] := b[l, t] = Module[{n = Length[l], s = Total[l]}, If[s == 0, 1, Sum[If[t != i && l[[i]] > If[i == n, 0, l[[i + 1]]], b[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, n}]]];
%t A214159 a[n_] := b[Table[3, n], 0];
%t A214159 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 29 2018, from first Maple program *)
%Y A214159 Column k=3 of A214021.
%K A214159 nonn
%O A214159 0,4
%A A214159 _Alois P. Heinz_, Jul 05 2012