cp's OEIS Frontend

A305122 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} (1+x^k)/(1-x^k).

%I A305122 #13 Jan 09 2025 19:08:46
%S A305122 0,0,1,2,4,8,16,28,47,78,126,198,306,464,694,1024,1490,2146,3061,4322,
%T A305122 6052,8408,11592,15872,21592,29192,39242,52468,69788,92376,121716,
%U A305122 159664,208569,271372,351732,454228,584546,749720,958472,1221560,1552210,1966698
%N A305122 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} (1+x^k)/(1-x^k).
%C A305122 Convolution of A305121 and A000009.
%C A305122 The g.f. Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >=  1} (1 + x ^k)/(1 - x^k) = Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 + x^k - 2*x^k) is congruent mod 2 to Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) = -G(-x^2), where G(x) is the g.f. of A112329. It follows that a(n) is odd iff n = 2*k^2 for some positive integer k. - _Peter Bala_, Jan 07 2025
%H A305122 Vaclav Kotesovec, <a href="/A305122/b305122.txt">Table of n, a(n) for n = 0..10000</a>
%F A305122 a(n) = A305101(n) - A305124(n).
%F A305122 a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).
%t A305122 nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A305122 Cf. A305102, A305101, A305104, A305105, A305124.
%Y A305122 Cf. A116680, A112329, A305121.
%K A305122 nonn
%O A305122 0,4
%A A305122 _Vaclav Kotesovec_, May 26 2018