A343204 Numerators of coefficients in expansion of Product_{k>=1} (1 + x^k)^(1/2).

1, 1, 3, 13, 67, 239, 1031, 2501, 36579, 109915, 468653, 1043851, 9395751, 21232827, 97493519, 235880373, 7717800611, 17385733651, 82456426833, 175398844079, 1578297716013, 3634938193489, 15867173716609, 34517119775523, 619312307079687, 1363237700933583

Offset: 0

Author

Ilya Gutkovskiy, Apr 07 2021

Keywords

• nonn
• frac

Examples

			1, 1/2, 3/8, 13/16, 67/128, 239/256, 1031/1024, 2501/2048, 36579/32768, 109915/65536, 468653/262144, 1043851/524288, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1631

Crossrefs

Cf. A000009, A022567, A046161 (denominators), A061159, A098987.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d/2, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 25; CoefficientList[Series[Product[(1 + x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator
a[n_] := a[n] = If[n == 0, 1, (1/(2 n)) Sum[Sum[Mod[d, 2] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}] // Numerator

Formula

a(n) / A046161(n) ~ exp(sqrt(n/6)*Pi) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Apr 12 2021