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A327739 Expansion of 1 / (1 - Sum_{i>=1} Sum_{j=1..i} x^(i*j)).

%I A327739 #8 Sep 27 2019 16:29:15
%S A327739 1,1,2,4,9,18,38,78,163,338,703,1458,3031,6293,13073,27150,56396,
%T A327739 117130,243289,505310,1049552,2179938,4527804,9404355,19533126,
%U A327739 40570816,84266725,175024267,363530253,755062265,1568285122,3257371187,6765649491,14052439669
%N A327739 Expansion of 1 / (1 - Sum_{i>=1} Sum_{j=1..i} x^(i*j)).
%C A327739 Invert transform of A038548.
%H A327739 Alois P. Heinz, <a href="/A327739/b327739.txt">Table of n, a(n) for n = 0..1000</a>
%F A327739 G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^k)).
%F A327739 a(0) = 1; a(n) = Sum_{k=1..n} A038548(k) * a(n-k).
%p A327739 a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
%p A327739       ceil(numtheory[sigma][0](i)/2), i=1..n))
%p A327739     end:
%p A327739 seq(a(n), n=0..34);  # _Alois P. Heinz_, Sep 23 2019
%t A327739 nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A327739 a[0] = 1; a[n_] := a[n] = Sum[Floor[(DivisorSigma[0, k] + 1)/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]
%Y A327739 Cf. A038548, A129921, A182269, A211856.
%K A327739 nonn
%O A327739 0,3
%A A327739 _Ilya Gutkovskiy_, Sep 23 2019