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A298321
The Nekrasov-Okounkov sequence.
1
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
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
Sequence in context: A329255 A262535 A096827 * A226142 A063826 A320120
KEYWORD
nonn
AUTHOR
Kenta Suzuki, Jan 17 2018
EXTENSIONS
More terms from Vincent Delecroix, Oct 05 2018
STATUS
approved