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A285472 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k)).

%I A285472 #12 Feb 03 2025 20:36:40
%S A285472 1,1,2,3,6,8,13,18,28,38,55,74,106,140,192,253,342,444,588,758,992,
%T A285472 1267,1634,2072,2650,3334,4218,5276,6627,8234,10262,12682,15708,19308,
%U A285472 23764,29070,35597,43340,52792,64008,77622,93724,113160,136124,163712,196225
%N A285472 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k)).
%C A285472 a(n) is the number of overpartitions wherein only parts that are a multiple of four may be overlined. - _Alois P. Heinz_, Feb 03 2025
%H A285472 Alois P. Heinz, <a href="/A285472/b285472.txt">Table of n, a(n) for n = 0..5000</a>
%F A285472 a(n) ~ sqrt(3) * exp(sqrt(3*n)*Pi/2) / (16*n).
%p A285472 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A285472       b(n, i-1)+`if`(irem(i, 4)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i)))
%p A285472     end:
%p A285472 a:= n-> b(n$2):
%p A285472 seq(a(n), n=0..45);  # _Alois P. Heinz_, Feb 03 2025
%t A285472 nmax = 50; CoefficientList[Series[Product[((1+x^(4*k))/(1-x^k)), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A285472 Cf. A006950, A015128, A279328, A266648, A285460.
%K A285472 nonn
%O A285472 0,3
%A A285472 _Vaclav Kotesovec_, Apr 19 2017