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%I #19 Jun 08 2015 06:00:26
%S 1,1,2,2,4,8,16,48,160,448,2048,6400,31232,125952,604160,3119104,
%T 15638528,93478912,550141952,3367698432,24049516544,146207539200,
%U 1203934593024,7615928598528,67190404415488,468355947429888,4196459066949632,33260378783744000
%N Sum of the squares of numbers of chess tableaux over all partitions of n.
%H Alois P. Heinz, <a href="/A108774/b108774.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.
%F a(n) = Sum_{lambda : partitions(n)} chess(lambda)^2, where chess(k) is the number of standard Young tableaux (SYT) with cell(i,j)+i+j == 1 mod 2. - _Alois P. Heinz_, Jun 30 2012
%p b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
%p add(`if`(irem(s+i-args[i], 2)=1 and args[i]>`if`(i=nargs, 0,
%p args[i+1]), b(subsop(i=args[i]-1, [args])[]), 0), i=1..nargs))
%p end:
%p g:= (n, i, l)-> `if`(n=0 or i=1, b(l[], 1$n)^2, `if`(i<1, 0,
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p a:= n-> `if`(n<2, 1, g(n, n, [])):
%p seq(a(n), n=0..27); # _Alois P. Heinz_, Jul 01 2012
%t b[args_List] := b[args] = Module[{s=Total[args], nargs=Length[args]}, If[s == 0, 1, Sum[If[Mod[s+i-args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i+1]] ], b[ReplacePart[args, i -> args[[i]]-1]], 0], {i, 1, nargs}] ] ]; g[n_, i_, l_List] := g[n, i, l] = If[n == 0 || i == 1, b[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}] ] ]; a[n_] := If[n<2, 1, g[n, n, {}]]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jun 08 2015, after _Alois P. Heinz_ *)
%Y Cf. A214087.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 27 2005
%E More terms from _Alois P. Heinz_, Jun 30 2012
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