A298321 The Nekrasov-Okounkov sequence.

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

1,5

Comments

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.

Links

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.

Example

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.

Mathematica

(* 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 *)

Prog

(Julia)
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
(PARI) {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

Crossrefs

Cf. A210590, A234937.

Sequence in context: A329255 A262535 A096827 * A226142 A063826 A320120

Adjacent sequences: A298318 A298319 A298320 * A298322 A298323 A298324

Keyword

nonn

Author

Kenta Suzuki, Jan 17 2018

Extensions

More terms from Vincent Delecroix, Oct 05 2018

Status

approved