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

1, 1, 1, 4, 5, 1, 18, 29, 12, 1, 120, 218, 119, 22, 1, 840, 1814, 1285, 345, 35, 1, 7920, 18144, 14674, 5205, 805, 51, 1, 75600, 196356, 185080, 79219, 16450, 1624, 70, 1, 887040, 2427312, 2515036, 1258628, 324569, 43568, 2954, 92, 1, 10886400, 32304240, 37012572, 21034376, 6431733, 1088409, 101178, 4974, 117, 1

Offset

0,4

Comments

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

Links

G. C. Greubel, Rows n=0..50 of triangle, flattened

Richard P. Stanley, Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions arXiv:0807.0383 [math.CO], 2009.

Formula

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

Example

Table starts as:
     1;
     1,     1;
     4,     5,     1;
    18,    29,    12,    1;
   120,   218,   119,   22,   1;
   840,  1814,  1285,  345,  35,  1;
  7920, 18144, 14674, 5205, 805, 51,  1;
  ...

Mathematica

w=9; MapIndexed[ CoefficientList[#1, t] Tr[#2-1]! &, CoefficientList[Series[Product[(1-x^i)^(-1-t), {i, w}], {x, 0, w}], x]];
or alternatively:
CoefficientList[#, t] & /@ Table[1/n! Tr[(NumberOfTableaux[#1]^2 Apply[Times, t + Flatten[hooklength[#1]]^2] &) /@ Partitions[n]], {n, 0, 9}]
or alternatively:
Table[1/n!Tr[NumberOfTableaux[#]^2 f[ Flatten[hooklength[#]]^2, e, k, n ]&/@ Partitions[n] ], {n, 0, 9}, {k, 0, n}]
with e and f defined as:
e[n_, v_]:= Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#]==n&]];
f[li_List, fun_, par_, k_]:=fun[par, k]/.Thread[Array[Subscript[x, #1]&, Length[li]]->li];

Crossrefs

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

T(2n,n) gives A338755.

Sequence in context: A369950 A266699 A234937 * A108446 A283263 A109962

Adjacent sequences: A210587 A210588 A210589 * A210591 A210592 A210593

Keyword

nonn,tabl

Author

Wouter Meeussen, Mar 24 2012

Status

approved