A160235 The maximal coefficient of (1+x)*(1+x^4)*(1+x^9)*...*(1+x^(n^2)).

1, 1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 14, 21, 32, 54, 87, 144, 230, 383, 671, 1158, 1981, 3408, 6246, 10925, 19463, 34624, 63941, 114954, 208429, 380130, 707194, 1298600, 2379842, 4398644, 8253618, 15303057, 28453809, 53091455, 100061278, 187446097

Offset: 0

Author

Theodore Kolokolnikov, May 05 2009

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..200 from Seiichi Manyama)
• Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Crossrefs

Cf. A025591, A033461, A359319, A359320, A369728.

Programs

Maple

for N from 1 to 40 do
p := expand(product(1+x^(n^2), n=1..N)):
L:=convert(PolynomialTools[CoefficientVector](p, x), list):
mmax := max(op(map(abs, L)));
lprint(mmax):
end:

Mathematica

p = 1; Table[p = Expand[p*(1 + x^(n^2))]; Max[CoefficientList[p, x]], {n, 1, 50}] (* Vaclav Kotesovec, May 04 2018 *)
nmax = 100; poly = ConstantArray[0, nmax*(nmax+1)*(2*nmax+1)/6 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, k*(k+1)*(2*k+1)/6, k^2, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 30 2022 *)

Formula

An asymptotic formula is a(n) ~ sqrt(10/Pi) * n^(-5/2) * 2^n. See for example the reference by Finch.

More precise asymptotics: a(n) ~ sqrt(10/Pi) * 2^n / n^(5/2) * (1 - 35/(18*n) + ...). - Vaclav Kotesovec, Dec 30 2022

Extensions

a(0)=1 prepended by Seiichi Manyama, Dec 26 2022