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A210590 Triangle of numbers generated by the Nekrasov-Okounkov formula. 6

%I

%S 1,1,1,4,5,1,18,29,12,1,120,218,119,22,1,840,1814,1285,345,35,1,7920,

%T 18144,14674,5205,805,51,1,75600,196356,185080,79219,16450,1624,70,1,

%U 887040,2427312,2515036,1258628,324569,43568,2954,92,1,10886400,32304240,37012572,21034376,6431733,1088409,101178,4974,117,1

%N Triangle of numbers generated by the Nekrasov-Okounkov formula.

%C Row sums are A000712, alternating sign row sums are zero (except for first row); application of the Nekrasov-Okounkov formula; see A138782.

%H G. C. Greubel, <a href="/A210590/b210590.txt">Rows n=0..50 of triangle, flattened</a>

%H 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 E.g.f.: Product_{i=1..n} (1 - x^i)^(-1 - t).

%e Table starts as:

%e 1;

%e 1, 1;

%e 4, 5, 1;

%e 18, 29, 12, 1;

%e 120, 218, 119, 22, 1;

%e 840, 1814, 1285, 345, 35, 1;

%e 7920, 18144, 14674, 5205, 805, 51, 1;

%e ...

%t w=9; MapIndexed[ CoefficientList[#1,t] Tr[#2-1]! &, CoefficientList[Series[Product[(1-x^i)^(-1-t), {i,w}], {x,0,w}], x]];

%t or alternatively:

%t CoefficientList[#, t] & /@ Table[1/n! Tr[(NumberOfTableaux[#1]^2 Apply[Times, t + Flatten[hooklength[#1]]^2] &) /@ Partitions[n]], {n,0,9}]

%t or alternatively:

%t Table[1/n!Tr[NumberOfTableaux[#]^2 f[ Flatten[hooklength[#]]^2,e,k,n ]&/@ Partitions[n] ],{n,0,9},{k,0,n}]

%t with e and f defined as:

%t e[n_,v_]:= Tr[Times @@@ Select[Subsets[Table[Subscript[x,j],{j,v}]],Length[#]==n&]];

%t f[li_List,fun_,par_,k_]:=fun[par,k]/.Thread[Array[Subscript[x,#1]&,Length[li]]->li];

%Y Cf. A000712, A053529, A057623, A138782, A234937.

%Y T(2n,n) gives A338755.

%K nonn,tabl

%O 0,4

%A _Wouter Meeussen_, Mar 24 2012

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Last modified May 7 10:32 EDT 2021. Contains 343650 sequences. (Running on oeis4.)