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A279328 Expansion of Product_{k>=1} (1 + x^(2*k)) / (1 - x^k).

%I A279328 #25 Feb 03 2025 20:46:31
%S A279328 1,1,3,4,8,11,20,27,44,60,92,124,183,244,348,461,640,840,1144,1488,
%T A279328 1992,2572,3393,4348,5668,7212,9301,11760,15024,18880,23924,29892,
%U A279328 37596,46728,58376,72193,89644,110340,136248,166968,205115,250316,306056,372032,452876
%N A279328 Expansion of Product_{k>=1} (1 + x^(2*k)) / (1 - x^k).
%H A279328 Alois P. Heinz, <a href="/A279328/b279328.txt">Table of n, a(n) for n = 0..5000</a> (first 1001 terms from Seiichi Manyama)
%H A279328 Aidan Carlson, Brian Hopkins, and James A. Sellers, <a href="https://doi.org/10.47443/dml.2024.130">Enumeration modulo four of overpartitions wherein only even parts may be overlined</a>, Disc. Math. Lett. (2024) Vol. 14, 95-102. See p. 96.
%F A279328 G.f.: Product_{k>=1} (1 + x^(2*k)) / (1 - x^k).
%F A279328 a(n) ~ sqrt(5/6) * exp(sqrt(5*n/6)*Pi) / (8*n). - _Vaclav Kotesovec_, Dec 10 2016
%e A279328 G.f.: 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 11*x^5 + 20*x^6 + 27*x^7 + 44*x^8 + ...
%p A279328 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A279328       b(n, i-1)+(2-irem(i, 2))*add(b(n-i*j, i-1), j=1..n/i)))
%p A279328     end:
%p A279328 a:= n-> b(n$2):
%p A279328 seq(a(n), n=0..44);  # _Alois P. Heinz_, Feb 03 2025
%t A279328 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k)) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 10 2016 *)
%Y A279328 Cf. Product_{k>=1} (1 + x^(m*k)) / (1 - x^k): A015128 (m=1), this sequence (m=2), A266648 (m=3).
%K A279328 nonn
%O A279328 0,3
%A A279328 _Seiichi Manyama_, Dec 09 2016