A234937 -id:A234937 - 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).

A338755 Central coefficients of number triangle A210590.

Original entry on oeis.org

1, 5, 119, 5205, 324569, 26519745, 2681170547, 323104570789, 45224035123553, 7211322045457101, 1290620989042420815, 256193650031596282005, 55863607060241676345961, 13273922770286234753307065, 3413846723448521483558054235, 944832714523233697801280445525, 280003865538498845896076940256065

Offset: 0

Seiichi Manyama, Nov 07 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A210590, A234937.

{a(n) = my(t='t); (2*n)!*polcoef(polcoef(prod(k=1, 2*n, (1-x^k+x*O(x^(2*n)))^(-1-t)), 2*n), n)}

{a(n) = my(t='t); if(n==0, 1, (2*n)!*polcoef(polcoef(exp(sum(k=1, 2*n, (1+t)*sigma(k)*(x^k+x*O(x^(2*n)))/k)), 2*n), n))}

a(n) = A210590(2*n, n) = (-1)^n * A234937(2*n, n).

A298321 The Nekrasov-Okounkov sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 3, 8, 6, 9, 8, 9, 12, 13, 11, 13, 12, 16, 18, 19, 18, 19, 21, 22, 22, 24, 24, 27, 25, 26, 29, 28, 31, 33, 32, 34, 32, 37, 35, 36, 37, 38, 42, 42, 41, 42, 43, 46, 48, 48, 45, 48, 50, 53, 54, 54, 51, 56, 56, 55, 58, 59, 60, 62, 62, 62

Offset: 1

Kenta Suzuki, Jan 17 2018

nonn

a(n) is the degree in terms of z of the coefficient of x^n's highest degree irreducible factor in Product_{m>=1} (1-x^m)^(z-1). This can be calculated by reducing the polynomial in the Nekrasov-Okounkov formula.

			For n = 5, a(n) = 2 because the coefficient of x^5 is Product_{m>=1} (1-x^m)^(z-1). This can be factorized as -(z-7)*(z-4)*(z-1)*(z^2 -23*z + 30)/120.

Vincent Delecroix, Table of n, a(n) for n = 1..506
Vincent Delecroix, SageMath script to generate the sequence
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, 60 no. 1 (2010), pp. 1-29.
Nikita A. Nekrasov and Andrei Okounkov, Seiberg-Witten Theory and Random Partitions, arXiv:hep-th/0306238, 2003.

Cf. A210590, A234937.

using Nemo
function A298321(len)
    R, z = PolynomialRing(ZZ, 'z')
    Q = [R(1)]; S = [1, 1, 1, 1]
    for n in 1:len-4
        p = z*sum(sigma(ZZ(k), 1)*risingfac(n-k+1, k-1)*Q[n-k+1] for k in 1:n)
        push!(Q, p)
        for (f, m) in factor(p)
            deg = degree(f)
            deg > 1 && push!(S, deg)
        end
    end
S end
A298321(72) |> println # Peter Luschny, Oct 27 2018, after Vincent Delecroix

(* This naive program is not suitable to compute a large number of terms *) a[n_] := a[n] = SeriesCoefficient[Product[(1-x^m)^(z-1), {m, 1, n}], {x, 0, n}] // Factor // Last // Exponent[#, z]&;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 18 2019 *)

{a(n) = vecmax(apply(x->poldegree(x), factor(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(z-1)), n))[, 1]))} \\ Seiichi Manyama, Nov 07 2020

More terms from Vincent Delecroix, Oct 05 2018

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

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A338755 Central coefficients of number triangle A210590.

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A298321 The Nekrasov-Okounkov sequence.

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