A350504 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^5) * ... * (1 + x^(2*n-1)).

1, 1, 1, 1, 2, 2, 3, 5, 8, 13, 22, 38, 68, 118, 211, 380, 692, 1262, 2316, 4277, 7930, 14745, 27517, 51541, 96792, 182182, 343711, 650095, 1231932, 2338706, 4447510, 8472697, 16164914, 30884150, 59086618, 113189168, 217091832, 416839177, 801247614, 1541726967, 2969432270

Offset: 0

Author

Ilya Gutkovskiy, Jan 28 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A025591, A039828, A160235, A350457.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1+x^(2*n-1))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 28 2022

Mathematica

b[n_] := b[n] = If[n == 0, 1, Expand[(1 + x^(2*n - 1))*b[n - 1]]];
a[n_] := Max[CoefficientList[b[n], x]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

PARI

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(2*k-1)))); \\ Seiichi Manyama, Jan 28 2021

Formula

a(n) ~ sqrt(3) * 2^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 04 2022