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A210590 Triangle of numbers generated by the Nekrasov-Okounkov formula. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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: A196848 A266699 A234937 * A108446 A283263 A109962

Adjacent sequences:  A210587 A210588 A210589 * A210591 A210592 A210593

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Mar 24 2012

STATUS

approved

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Last modified June 21 08:56 EDT 2021. Contains 345358 sequences. (Running on oeis4.)