A108774 Sum of the squares of numbers of chess tableaux over all partitions of n.

1, 1, 2, 2, 4, 8, 16, 48, 160, 448, 2048, 6400, 31232, 125952, 604160, 3119104, 15638528, 93478912, 550141952, 3367698432, 24049516544, 146207539200, 1203934593024, 7615928598528, 67190404415488, 468355947429888, 4196459066949632, 33260378783744000

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.

Formula

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

Maple

b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
      add(`if`(irem(s+i-args[i], 2)=1 and args[i]>`if`(i=nargs, 0,
      args[i+1]), b(subsop(i=args[i]-1, [args])[]), 0), i=1..nargs))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, b(l[], 1$n)^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> `if`(n<2, 1, g(n, n, [])):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 01 2012

Mathematica

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 *)

Crossrefs

Cf. A214087.

Sequence in context: A001137 A123593 A122748 * A063402 A175195 A369289

Adjacent sequences: A108771 A108772 A108773 * A108775 A108776 A108777

Keyword

nonn

Author

N. J. A. Sloane, Jun 27 2005

Extensions

More terms from Alois P. Heinz, Jun 30 2012

Status

approved