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
(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
KEYWORD
nonn
AUTHOR
Kenta Suzuki, Jan 17 2018
EXTENSIONS
More terms from Vincent Delecroix, Oct 05 2018
STATUS
approved