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A210590 Triangle of numbers generated by the Nekrasov-Okounkov formula.

%I A210590 #39 Nov 07 2020 11:00:30
%S A210590 1,1,1,4,5,1,18,29,12,1,120,218,119,22,1,840,1814,1285,345,35,1,7920,
%T A210590 18144,14674,5205,805,51,1,75600,196356,185080,79219,16450,1624,70,1,
%U A210590 887040,2427312,2515036,1258628,324569,43568,2954,92,1,10886400,32304240,37012572,21034376,6431733,1088409,101178,4974,117,1
%N A210590 Triangle of numbers generated by the Nekrasov-Okounkov formula.
%C A210590 Row sums are A000712, alternating sign row sums are zero (except for first row); application of the Nekrasov-Okounkov formula; see A138782.
%H A210590 G. C. Greubel, <a href="/A210590/b210590.txt">Rows n=0..50 of triangle, flattened</a>
%H A210590 Richard P. Stanley, <a href="http://arxiv.org/abs/0807.0383">Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions</a>  arXiv:0807.0383 [math.CO], 2009.
%F A210590 E.g.f.: Product_{i=1..n} (1 - x^i)^(-1 - t).
%e A210590 Table starts as:
%e A210590      1;
%e A210590      1,     1;
%e A210590      4,     5,     1;
%e A210590     18,    29,    12,    1;
%e A210590    120,   218,   119,   22,   1;
%e A210590    840,  1814,  1285,  345,  35,  1;
%e A210590   7920, 18144, 14674, 5205, 805, 51,  1;
%e A210590   ...
%t A210590 w=9; MapIndexed[ CoefficientList[#1,t] Tr[#2-1]! &, CoefficientList[Series[Product[(1-x^i)^(-1-t), {i,w}], {x,0,w}], x]];
%t A210590 or alternatively:
%t A210590 CoefficientList[#, t] & /@ Table[1/n! Tr[(NumberOfTableaux[#1]^2 Apply[Times, t + Flatten[hooklength[#1]]^2] &) /@ Partitions[n]], {n,0,9}]
%t A210590 or alternatively:
%t A210590 Table[1/n!Tr[NumberOfTableaux[#]^2 f[ Flatten[hooklength[#]]^2,e,k,n ]&/@ Partitions[n] ],{n,0,9},{k,0,n}]
%t A210590 with e and f defined as:
%t A210590 e[n_,v_]:= Tr[Times @@@ Select[Subsets[Table[Subscript[x,j],{j,v}]],Length[#]==n&]];
%t A210590 f[li_List,fun_,par_,k_]:=fun[par,k]/.Thread[Array[Subscript[x,#1]&,Length[li]]->li];
%Y A210590 Cf. A000712, A053529, A057623, A138782, A234937.
%Y A210590 T(2n,n) gives A338755.
%K A210590 nonn,tabl
%O A210590 0,4
%A A210590 _Wouter Meeussen_, Mar 24 2012