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 A108774 Sum of the squares of numbers of chess tableaux over all partitions of n. 5

%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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Last modified August 6 15:31 EDT 2024. Contains 374974 sequences. (Running on oeis4.)