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A214087 Sum of the squares of numbers of nonconsecutive tableaux over all partitions of n.

%I A214087 #17 May 23 2018 11:00:57
%S A214087 1,1,1,2,6,21,92,489,3000,20970,166714,1467337,14212491,149992662,
%T A214087 1723338952,21393028409,285061374438,4054622024814,61301381208116,
%U A214087 982904573560309,16672187358390360,298389960090957330,5617735345244596804,110942937545014894799
%N A214087 Sum of the squares of numbers of nonconsecutive tableaux over all partitions of n.
%C A214087 A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.
%H A214087 Alois P. Heinz, <a href="/A214087/b214087.txt">Table of n, a(n) for n = 0..40</a>
%H A214087 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 A214087 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A214087 b:= proc(l, t) option remember; local n, s; n, s:= nops(l),
%p A214087        add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and l[i]>
%p A214087       `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p A214087     end:
%p A214087 g:= (n, i, l)-> `if`(n=0 or i=1, b([l[], 1$n], 0)^2, `if`(i<1, 0,
%p A214087                  add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p A214087 a:= n-> `if`(n<2, 1, g(n, n, [])):
%p A214087 seq(a(n), n=0..20);
%t A214087 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 A214087 g[n_, i_, l_] := If[n == 0 || i == 1, b[Join[l, Table[1, n]], 0]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Table[i, j]]], {j, 0, n/i}]]];
%t A214087 a[n_] := If[n < 2, 1, g[n, n, {}]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 23 2018, translated from Maple *)
%Y A214087 Cf. A108774, A237770.
%K A214087 nonn
%O A214087 0,4
%A A214087 _Alois P. Heinz_, Jul 02 2012