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

%I A343204 #14 Jun 25 2025 20:12:51
%S A343204 1,1,3,13,67,239,1031,2501,36579,109915,468653,1043851,9395751,
%T A343204 21232827,97493519,235880373,7717800611,17385733651,82456426833,
%U A343204 175398844079,1578297716013,3634938193489,15867173716609,34517119775523,619312307079687,1363237700933583
%N A343204 Numerators of coefficients in expansion of Product_{k>=1} (1 + x^k)^(1/2).
%H A343204 Alois P. Heinz, <a href="/A343204/b343204.txt">Table of n, a(n) for n = 0..1631</a>
%F A343204 a(n) / A046161(n) ~ exp(sqrt(n/6)*Pi) / (4*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Apr 12 2021
%e A343204 1, 1/2, 3/8, 13/16, 67/128, 239/256, 1031/1024, 2501/2048, 36579/32768, 109915/65536, 468653/262144, 1043851/524288, ...
%p A343204 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A343204      `if`(d::odd, d/2, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A343204     end:
%p A343204 a:= n-> numer(b(n)):
%p A343204 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 12 2021
%t A343204 nmax = 25; CoefficientList[Series[Product[(1 + x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator
%t A343204 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
%Y A343204 Cf. A000009, A022567, A046161 (denominators), A061159, A098987.
%K A343204 nonn,frac
%O A343204 0,3
%A A343204 _Ilya Gutkovskiy_, Apr 07 2021