A138782 -id:A138782 - cp's OEIS frontend

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

Wouter Meeussen, Mar 24 2012

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

			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;
  ...

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.

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

T(2n,n) gives A338755.

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];

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

A138784 a(n) = n(n-1)(n-2)(27n^3 - 174n^2 + 511n - 552)n!/48.

Original entry on oeis.org

0, 0, 108, 5232, 154200, 3747600, 82908000, 1756661760, 36715472640, 770648256000, 16426162368000, 358029745920000, 8015066233574400, 184788598259865600, 4394609711331840000, 107899664364011520000

Offset: 1

Emeric Deutsch, May 15 2008

a(n) = Sum [f(L)^2 Sum h(u)^2*h(v)^2h(w)^2], where L is a partition of n, f(L) is the number of standard Young tableaux of shape L, h(z) is the hook length of the box z in L (i.e., in the Ferrers diagram of L), the inner summation is over all unordered triples of distinct boxes u, v and w in L and the outer summation is over all partitions of n. Example: a(3)=108 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the values of f(L) are 1, 2, 1, respectively, the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, Sum h(u)^2*h(v)^2*h(w)^2 = 36, 9, 36, respectively and now a(n) 1^2*36 + 2^2*9 + 1^2*36 = 108.

In Proposition 6.12 of the Han paper the number 600 should be replaced by 552. - Emeric Deutsch, Dec 07 2015

Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 29).

Cf. A138782, A138783.

[n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*Factorial(n)/48: n in [1..25]]; // Vincenzo Librandi, Dec 08 2015

A138784 := proc(n)
    n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*factorial(n)/48 ;
end proc:
seq(A138784(n),n=1..30) ;

Table[n (n - 1) (n - 2) (27 n^3 - 174 n^2 + 511 n - 552) n!/48, {n, 25}] (* Vincenzo Librandi, Dec 08 2015 *)

a(n) = n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*n!/48; \\ Altug Alkan, Dec 07 2015

D-finite 9*(n-3) *(4179399*n -20778487) *a(n) +(-37614591*n^3 -15080676*n^2 +317158871*n +3150537632) *a(n-1) +2*(44621643*n +200235296) *(-1+n)^2*a(n-2)=0. - R. J. Mathar, Dec 10 2015

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

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A138784 a(n) = n(n-1)(n-2)(27n^3 - 174n^2 + 511n - 552)n!/48.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A138784 a(n) = n*(n-1)*(n-2)(27*n^3 - 174*n^2 + 511*n - 552)*n!/48.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A138784 a(n) = n(n-1)(n-2)(27n^3 - 174n^2 + 511n - 552)n!/48.