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%I #19 Jul 19 2017 15:48:40
%S 1,1,1,1,2,2,4,7,16,37,107,282,1020,2879,12507,39347,179231,687974,
%T 3225246,14955561,75999551,392585613,2271201137,12183159188,
%U 81562521256,446611878413,3336304592155,19202329389234,152803821604669,958953289839930,7835058287650579
%N Sum of the squares of numbers of nonconsecutive chess tableaux over all partitions of n.
%C A standard Young tableau (SYT) with cell(i,j)+i+j == 1 mod 2 for all cells where entries m and m+1 never appear in the same row is called a nonconsecutive chess tableau.
%H Alois P. Heinz, <a href="/A238184/b238184.txt">Table of n, a(n) for n = 0..50</a>
%H 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 Jonas Sjöstrand, <a href="https://arxiv.org/abs/math/0309231v3">On the sign-imbalance of partition shapes</a>, arXiv:math/0309231v3 [math.CO], 2005.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F a(n) = Sum_{lambda : partitions(n)} ncc(lambda)^2, where ncc(k) is the number of nonconsecutive chess tableaux of shape k.
%e a(7) = 1 + 2^2 + 1 + 1 = 7:
%e .
%e : [1111111] : [22111] : [3211] : [322] : <- shapes
%e :-----------+--------------+---------+---------:
%e : [1] : [1 6] [1 4] : [1 4 7] : [1 4 7] :
%e : [2] : [2 7] [2 5] : [2 5] : [2 5] :
%e : [3] : [3] [3] : [3] : [3 6] :
%e : [4] : [4] [6] : [6] : :
%e : [5] : [5] [7] : : :
%e : [6] : : : :
%e : [7] : : : :
%p b:= proc(l, t) option remember; local n, s;
%p n, s:= nops(l), add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and
%p irem(s+i-l[i], 2)=1 and l[i]>`if`(i=n, 0, l[i+1]), b(subsop(
%p i=`if`(i=n and l[n]=1, [][], l[i]-1), l), i), 0), i=1..n))
%p end:
%p g:= (n, i, l)-> `if`(n=0 or i=1, b([l[], 1$n], 0)^2, `if`(i<1, 0,
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p a:= n-> g(n, n, []):
%p seq(a(n), n=0..32);
%t b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Total[l]}; If[s == 0, 1, Sum[If[t != i && Mod[s+i-l[[i]], 2] == 1 && l[[i]] > If[i==n, 0, l[[i+1]]], b[ReplacePart[l, i -> If[i==n && l[[n]]==1, Nothing, l[[i]]-1]], i], 0], {i, 1, n}]]]; g[n_, i_, l_] := g[n, i, l] = If[n==0 || i==1, b[Join[l, Array[1&, n]], 0]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Feb 17 2017, translated from Maple *)
%Y Cf. A108774, A214088, A214459, A214460, A214461, A238020.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Feb 19 2014
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