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Expansion of 1 / (1 - Sum_{i>=1} Sum_{j=1..i} x^(i*j)).
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%I #8 Sep 27 2019 16:29:15

%S 1,1,2,4,9,18,38,78,163,338,703,1458,3031,6293,13073,27150,56396,

%T 117130,243289,505310,1049552,2179938,4527804,9404355,19533126,

%U 40570816,84266725,175024267,363530253,755062265,1568285122,3257371187,6765649491,14052439669

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

%C Invert transform of A038548.

%H Alois P. Heinz, <a href="/A327739/b327739.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^k)).

%F a(0) = 1; a(n) = Sum_{k=1..n} A038548(k) * a(n-k).

%p a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*

%p ceil(numtheory[sigma][0](i)/2), i=1..n))

%p end:

%p seq(a(n), n=0..34); # _Alois P. Heinz_, Sep 23 2019

%t nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%t 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 Cf. A038548, A129921, A182269, A211856.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 23 2019