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%I #7 Feb 23 2018 18:53:38
%S 1,0,2,3,13,32,121,376,1406,5030,19632,76334,314582,1308550,5667494,
%T 24940458,113239394,523149560,2480434938,11968944532,59051754824
%N Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions non-singleton skew-partitions.
%C A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.
%e The a(4) = 13 tableaux:
%e 1 1 2 2 1 1 1 1
%e .
%e 1 2 2 1 1 2 1 1 1
%e 1 2 1
%e .
%e 1 2 1 1 1 1
%e 1 2 2 2 1 1
%e .
%e 1 2 1 1 1 1
%e 1 2 1
%e 2 2 1
%e .
%e 1 1
%e 1 1
%e 2 1
%e 2 1
%t undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
%t ehn[y_]:=ehn[y]=If[Total[y]=!=1,1,0]+Sum[ehn[c],{c,Select[undptns[y],Total[#]>1&&Total[y]-Total[#]>1&]}];
%t Table[Sum[ehn[y],{y,IntegerPartitions[n]}],{n,15}]
%Y Cf. A000085, A063834, A138178, A153452, A238690, A296188, A297388, A299925, A299926, A299966.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Feb 22 2018
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