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%I #15 Feb 23 2018 22:14:03
%S 1,4,14,60,252,1212,5880,30904,166976,952456,5587840,34217216,
%T 215204960,1401551376,9360467760,64384034784,453328282624,
%U 3274696185568,24173219998912,182546586425408
%N a(n) is the number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions skew partitions.
%C If y is an integer partition of n, 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(3) = 14 tableaux:
%e 1 2 3 1 2 2 1 1 2 1 1 1
%e .
%e 1 3 1 2 1 2 1 2 1 1 1 1
%e 2 3 2 1 2 1
%e .
%e 1 1 1 1
%e 2 2 1 1
%e 3 2 2 1
%t undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
%t chn[y_]:=Join[{{{},y}},Join@@Function[c,Append[#,y]&/@chn[c]]/@Take[undptns[y],{2,-2}]];
%t Table[Sum[Length[chn[y]],{y,IntegerPartitions[n]}],{n,8}]
%Y Cf. A000085, A063834, A138178, A153452, A296188, A296561, A297388, A299699, A299925.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Feb 21 2018
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