cp's OEIS Frontend

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

%I A160235 #34 Jan 31 2024 08:27:55
%S A160235 1,1,1,1,1,2,2,3,5,6,9,14,21,32,54,87,144,230,383,671,1158,1981,3408,
%T A160235 6246,10925,19463,34624,63941,114954,208429,380130,707194,1298600,
%U A160235 2379842,4398644,8253618,15303057,28453809,53091455,100061278,187446097
%N A160235 The maximal coefficient of (1+x)*(1+x^4)*(1+x^9)*...*(1+x^(n^2)).
%H A160235 Vaclav Kotesovec, <a href="/A160235/b160235.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Seiichi Manyama)
%H A160235 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 A160235 An asymptotic formula is a(n) ~ sqrt(10/Pi) * n^(-5/2) * 2^n. See for example the reference by Finch.
%F A160235 More precise asymptotics: a(n) ~ sqrt(10/Pi) * 2^n / n^(5/2) * (1 - 35/(18*n) + ...). - _Vaclav Kotesovec_, Dec 30 2022
%p A160235 for N from 1 to 40 do
%p A160235 p := expand(product(1+x^(n^2), n=1..N)):
%p A160235 L:=convert(PolynomialTools[CoefficientVector](p, x), list):
%p A160235 mmax := max(op(map(abs, L)));
%p A160235 lprint(mmax):
%p A160235 end:
%t A160235 p = 1; Table[p = Expand[p*(1 + x^(n^2))]; Max[CoefficientList[p, x]], {n, 1, 50}] (* _Vaclav Kotesovec_, May 04 2018 *)
%t A160235 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 A160235 Cf. A025591, A033461, A359319, A359320, A369728.
%K A160235 nonn
%O A160235 0,6
%A A160235 _Theodore Kolokolnikov_, May 05 2009
%E A160235 a(0)=1 prepended by _Seiichi Manyama_, Dec 26 2022