A160235 - OEIS

%I #34 Jan 31 2024 08:27:55

%S 1,1,1,1,1,2,2,3,5,6,9,14,21,32,54,87,144,230,383,671,1158,1981,3408,

%T 6246,10925,19463,34624,63941,114954,208429,380130,707194,1298600,

%U 2379842,4398644,8253618,15303057,28453809,53091455,100061278,187446097

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

%H Vaclav Kotesovec, <a href="/A160235/b160235.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Seiichi Manyama)

%H Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]

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

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

%p for N from 1 to 40 do

%p p := expand(product(1+x^(n^2), n=1..N)):

%p L:=convert(PolynomialTools[CoefficientVector](p, x), list):

%p mmax := max(op(map(abs, L)));

%p lprint(mmax):

%p end:

%t p = 1; Table[p = Expand[p*(1 + x^(n^2))]; Max[CoefficientList[p, x]], {n, 1, 50}] (* _Vaclav Kotesovec_, May 04 2018 *)

%t 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 *)

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

%K nonn

%O 0,6

%A _Theodore Kolokolnikov_, May 05 2009

%E a(0)=1 prepended by _Seiichi Manyama_, Dec 26 2022